a = 1, b = 6, c = 8 f(x) = x2 6x + 8 GeoGebra is the way to go, I believe. Find the Equation of a Quadratic (Parabola) Given 3 Points and then use the point-slope form to write the equation of the line. Trending Posts. Gain more insight into the quadratic formula and how it is used in quadratic equations. This math tutorial shows how to find a vertex form of a quadratic equation as well as the quadratic form from 2 points on a parabola. Here is a quadratic that willnotfactor: x27x3=0{x}^{2}-7x-3=0x27x3=0. @Adam: This would be a good question for the IntMath Forum. and then you must solve the system Again, thank you so much for putting together this wonderful page for people like me. (This gives the blue parabola as shown below). So I would get y= 2^2 + 2b + 3). Get better grades with tutoring from top-rated private tutors. My website is in the very same niche as yours and my users would definitely benefit from some of the information you present here. find the equation of the tangents to a quadratic curve through the point $(-7,1)$ 1 Conic sections ( coordinates of the point of intersection of tangents with the curve ) Thanks. First we factor the equation. Math is the study of numbers, space, and structure. But once again, we are not even trying to find an "x". This simplifies to a = 1. That is, y = ax + bx + c y = ax + bx + c From these we obtain What to do? The best answers are voted up and rise to the top, Not the answer you're looking for? How do you find exact values for the sine of all angles? The numerals a, b, and c are coefficients of the equation, and they represent known numbers. the ones I'm having trouble with are ones like Does the above equation represent a parabola? i find just a little problem solving a problem. If you're looking for detailed, step-by-step answers, you've come to the right place. If we use y = a(x h)2 + k, we can see from the graph that h = 1 and k = 0. I'm having trouble in determining the equation from its graph (>.<) The image only has 5 units for each positive and negative x and y. Substitute the vertex's coordinates for h and k in the vertex form. Then use a different method to check your work. find the "=0" points; in between the "=0" points, are intervals that are either greater than zero (>0), or; Can I use excel and choose polynomial and order 4? The equation of the parabola is y = ax2 + bx + c, where a can never equal zero. This online calculator is a quadratic equation solver that will solve a second-order polynomial equation such as ax 2 + bx + c = 0 for x, where a 0, using the quadratic formula. Thanks pal really helped me. 2 02 Quadratic Equations. 1) Find Quadratic Equation from 2 Points In order to find a quadratic equation from a graph using only 2 points, one of those points must be the vertex. There are multiple ways to reduce stress, including exercise, relaxation techniques, and healthy coping mechanisms. (Bookmark the Link Below)http://www.mariosmathtutoring.com/free-math-videos.html A quadratic equation has two solutions if the discriminant b^2 - 4ac is positive. x = [- (-7) ( (-7) 2 - 4 (1) (10))] / (2 (1)) = [ 7 (49 - 40) ] / 2 = [ 7 (9) ] / 2 = [ 7 3 ] / 2 And don't forget the parabolas in the "legs down" orientation: So how do we find the correct quadratic function for our original question (the one in blue)? I can solve any mathematic question you give me. Surround your math with. Given just 2 points, to find a linear equation, this is the formula: Example Problems - Quadratic Equations Example 1 - solve with quadratic formula Example 2 - solve using Indian method Example 3 - solve by factoring Example 4 - completing the square Example 5 - create quadratic ALL Example Problems - Work Rate Problems Example 3 - time to wash cars Example 4 - Excel Linear Programming Please reply soon. Looking for an introduction to parabolas? @ABC: Every parabola passes through at least one of the axes. Substituting 3 for x and 8 for y in, Solve the equation for a. Can the Spiritual Weapon spell be used as cover? Direct link to almadugomez's post how is the quadratic form, Posted 7 years ago. If you're looking for fast answers, you've come to the right place. Finding Roots of Quadratic Equation by Quadratic Formula Find a, b, and c values by comparing the given equation with ax 2 + bx + c = 0. Any further help I'll greatly appreciate it. Step 1: Enter the equation you want to solve using the quadratic formula. If the graph of y = f(x) = ax^2 + bx + c passes through (1,0) and (3,0) this means that f(1) = 0 and f(3) = 0. Vertex point: (|) Further point: (|) Computing a quadratic function out of three points Enter three points. Solve for b. Use the quadratic formula to check factoring, for instance. Step 2: Pick a point on the graph, and Writing the equation of a quadratic function given its graph . Just go about it the same as I did int he article: start with y = ax^2 + bx + c and substitute in your 3 points, then solve. He says that to graph a parabola you need to find the mirror point symmetrical to the Y-intercept. If you need help with your math homework, there are online calculators that can assist you. But what I'd also hope you'd consider is spending some of your time as a volunteer with students who don't have the means to afford math tutoring, but desperately need it. Consider a quadratic equation in standard form: You may also see the standard form called a general quadratic equation, or the general form. The equation is y=4xsquare-4x+4. how to graph a parabola ? How to Write Quadratic Equations Given a Vertex & Point We can use the vertex form to find a parabola's equation. I will carry this information with me until I forget it, which undoubtably will be very soon, in which case I will soon be back. Enter the vertex point and another point on the graph. This gives us y = a(x 1)2. If you're ever stuck on a math question, be sure to ask your teacher or a friend for clarification. One of the activities in my "Blue Meanies" game (at http://qpr.ca/math/applets/meanies/ )asks students to "guess" the equation of a parabola through three points by imagining the curve and using its geometry (in various ways) to determine the equation. If you preorder a special airline meal (e.g. Conic Sections: Parabola and Focus. Look no further our experts are here to help. example It is important that you know how to find solutions for quadratic equations using the quadratic formula. Direct link to Bentley S.'s post Im here, Posted 5 years ago. what if the curve has three x-intecepts? @haha not my real name: I'm not surprised this ends up in a loop. Hope it helps. But on my math homework, I we are working with conic sections and parabolas. Calculate a quadratic function given the vertex point Enter the vertex point and another point on the graph. Mathematics is the study of patterns and relationships between numbers, shapes, and other mathematical concepts. GeoGebra was not so useful for this task. There can be 0, 1 or 2 solutions to a quadratic equation. The y -intercept is (0, -3). In this tutorial, you'll see how to solve such a system by combining the equations together in a way so that one of the variables is eliminated. I did some digging and found a GeoGebra applet (no longer available) which draws a parabola through 3 points. In your example at the top of this page, you end up with the equation (#1), y= x^2+x-2 for the parabola but you rule it out because this equations leads to a y intercept of -2 whereas the graph shows a y intercept of -3. Possible answer: Do I substitute in the value of x and y (in this case x=2, y=3) into the equation y = x^2 +bx + c to get 3 = 2^ + 2b + c (although I thought c would be 3 as this is the point where the curve crosses the y-axis? If the graph of y = f(x) = ax^2 + bx + c passes through (1,0) and (3,0) this means that f(1) = 0 and f(3) = 0. Very well explained, I understood it well after reading it through once. Finding the equation of a parabola given certain data points is a worthwhile skill in mathematics. . Maybe someone who reads this could invent one? Why do many companies reject expired SSL certificates as bugs in bug bounties? Graphing calculators will probablynotbe equal to the precision of the quadratic formula. The quadratic formula gives solutions to the quadratic equation ax^2+bx+c=0 and is written in the form of x = (-b (b^2 - 4ac)) / (2a). Now you can also solve a quadratic equation through factoring, completing the square, or graphing, so why do we need the formula? The quadratic formula is used to solve quadratic equations. Substitute the second ordered pair and the value of a into the general equation. (parabola Legs towards West direction). @Evogod: Feel free to ask more about this in the IntMath Forum. In solving quadratics, you help yourself by knowing multiple ways to solve any equation. Part of Maths Algebraic skills Revise Test 1 2 Identify. Where does the word "Quadratic" come from? Instead of x, you can also write x^2. To find the quadratic functions whose graphs contain the points and we can evaluate at 1 and 0 to find Solving the first equation for gives . For example, (1, 5), (2,11) and (3,19). Let's start with the simplest case. I am a 41 year old who is about to study maths and physics at uni for the first time; stuff like this is fantastic. And thanks for sharing "Meanies"! Best math calculator ever seen. Further point: @Mick: Thanks for the positive feedback. That is, we can do it with software or without. It is deri, Posted 9 years ago. I have divided both sides by 6 to give me on the left. vegan) just to try it, does this inconvenience the caterers and staff? Then, see how find the value of that variable and use it to find the value of the other variable. For your second question, see also the 2 links I gave in my reply to Leah, above. Or a logarithmic graph, or asymptotic graph if all you have is the graph itself? I'm glad your found it useful! Often we have a set of data points from observations in an experiment, say, but we don't know the function that passes through our data points. If you have a general quadratic equation like this: ax^2+bx+c=0 ax2 + bx + c = 0 Then the formula will help you find the roots of a quadratic equation, i.e. First note, Posted 9 years ago. If you're seeing this message, it means we're having trouble loading external resources on our website. With a simple review of your work, you can find ways to improve and understand How to find quadratic equation from two points! Which "x" are you trying to calculate? Note: We could also make use of the fact that the x-value of the vertex of the parabola y = ax2 + bx + c is given by: Here's an example where there is no x-intercept. I've added 3 or 4 statements about axis of symmetry on this page to help you: Ive got a question, (sorry for my bad English). How Quadratic Regression Calculator Works? if you mean find the solution, yes, you would get -3 and 1. And it's still a nice day! let the equation of the quadratic function be: y = a x 2 + b x + c. since the curve passes through the points ( 3, 0), ( 2, 0), and ( 0, 30) . Another option could be to approach an existing entity that's doing interesting things (New York's Museum of Mathematics comes to mind) and offer your services as a researcher. Quadratic equation standard form. Explanation: Because the question specifies a function, we must discard the form that is not a function: x = ay2 +by + c and use only the form: y = ax2 +bx +c [1] Substitute the values of a, b and c into the general quadratic equation. It is neatly listed in order from the top down and was easy to follow. Work through it. instead of the formula, my textbook wants me to use factorization..how to i do x^2+2x-3=0? Example 1: Vertex form Graph the equation. There is simply no way to make an analogous equation for any polynomial of degree y for y>4, not enough operations are defined by the rules of mathematics. In order to find a quadratic equation from a graph, there are two simple methods one can employ: using 2 points, or using 3 points. Then apply the quadratic formula. The quadratic formula x=\dfrac {-b\pm\sqrt {b^2-4ac}} {2a} x = 2ab b2 4ac It may look a little scary, but you'll get used to it quickly! Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. []. How to Find a Quadratic Equation from a Graph: Step 1: Identify Points Step 2: Sub Points Into Vertex Form and Solve for a Step 3: Write Out Quadratic . Writing Quadratic Equations. It is a great app with all necessary tools for solving any maths problem. Substitute the third ordered pair and the values of a and b into the general equation. Finding a quadratic function with a parabola. when the only given is the equation?? Finding A Quadratic Equation From 2 Points On Parabola And The Vertex You. I do not enjoy math and I need some help. When a quadratic equation is graphed, it forms a parabola. We will now prepare a table for the roots of "X" which are "x1" and "x2", and . In the end across a set of locations I have values in the following form. By the way, do you know any college that has a doctorate in Mathematics on line as I have nothing else to do. Direct link to Cian Knight's post Where does the word "Quad, Posted 7 years ago. They can be used to calculate areas, formulate the speed of an object, and even to determine a product's profit. Two cloud shapes down and one to go. $$y_1=ax_1^2+bx_1+c$$ The quadratic formula gives solutions to the quadratic equation ax^2+bx+c=0 and is written in the form of x = (-b (b^2 - 4ac)) / (2a) Does any quadratic equation have two solutions? In this example, substituting a into, Square the expression inside the parentheses, multiply the terms by a's value and combine like terms to convert the equation to standard form. Why is there a voltage on my HDMI and coaxial cables? When using the quadratic formula, you must be attentive to the smallest details. I mean I have heard of so called Octic Equations which are of the form: In 1827, a mathematician by the last name of Abel proved that there is no way to make an analogous equation past the 4th degree. Whether you need help solving quadratic equations, inspiration for the upcoming science fair or the latest update on a major storm, Sciencing is here to help. Direct link to MBlackwll's post Hopefully this proof help, Posted 7 years ago. (All parabolas with axis parallel to the y-axis pass through the y-axis). I hope it makes more sense now. and i need to form a quadratic equation based on that could you pls help me out with it.. Hello Abhishek. Hey all, I can't believe I have to scan my math problem just to get it checked. I have tried hard but found none. 3/9/10 5:45 AM. Can you help me understand what he means? Point A (|) Point B (|) Point C (|) $$y\ =\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}x+\frac{\left(y_1+y_2-\left(x_1\left(\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}\right)+x_2\left(\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}\right)\right)\right)}{2}$$ $$y_3=ax_3^2+bx_3+c$$ for $a,b,c$, Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Use the calculator to verify the rounded results, but expect them to be slightly different. The equation of a a quadratic function can be determined from a graph showing the y-intecept, axis of symmetry and turn point. Which line is the last line that you multiply by 2 and which line do you add that too also how did you get the final answer of 1.5 for A and B? You may also see the standard form called a general quadratic equation, or the general form. First, let p(x) = ax^2+ b*x+ c. The derivative is p'(x) = 2a*x+b, so the maximum value of p occurs at the solution z of 0 = p. In order to find a quadratic equation from a graph, there are two simple methods one can employ: using 2 points, or using 3 points. $$L(x) = {y_1}\cdot{x - x_2 \over x_1 - x_2}\cdot{x - x_3 \over x_1 - x_3}+ {y_2}\cdot{x - x_1 \over x_2 - x_1}\cdot{x - x_3 \over x_2 - x_3}+ {y_3}\cdot{x - x_1 \over x_3 - x_1}\cdot{x - x_2 \over x_3 - x_2}$$, you can make the ansatz $$y=a^2+bx+c$$ and plug in all your points in this equation Given just 2 points, to find a linear equation, this is the formula: $$y\ =\frac {\left (y_2-y_1\right)} {\left (x_2-x_1\right)}x+\frac {\left (y_1+y_2-\left (x_1\left (\frac {\left (y_2-y_1\right)} {\left (x_2-x_1\right)}\right)+x_2\left (\frac {\left (y_2-y_1\right)} {\left (x_2-x_1\right)}\right)\right)\right)} {2}$$ In the same form of I appreciate the simple images to go along with the explanations, that also helped a lot. Check out my Huge ACT Math Video Course and my Huge SAT Math Video Course for sale athttp://mariosmathtutoring.teachable.comFor online 1-to-1 tutoring or more information about me see my website at:http://www.mariosmathtutoring.com* Organized List of My Video Lessons to Help You Raise Your Scores \u0026 Pass Your Class. Just as a quadratic equation can map a parabola, the parabola's points can help write a corresponding quadratic equation. Thanks. where the curve crosses the x-axis. Instead of x, you can also write x^2. In the same form of $y=$, what is the formula for a quadratic equation with 3 points? Replacing broken pins/legs on a DIP IC package, Styling contours by colour and by line thickness in QGIS, Identify those arcade games from a 1983 Brazilian music video. They'll just say it's simplified and only occasionally tell you the math rule that was used but I believe that is because I'm on trial and not paying. Direct link to Patrick's post For the quadratic formula, Posted 7 years ago. What is the relationship between three points on a quadratic curve and the curves coefficients? Solve mathematic question I can solve any mathematic question you give me. If the discriminant is positive there are two solutions, if negative there is no solution, if equlas 0 there is 1 solution. Where are we getting the 2 from and also why would we add it to the second line? @Ethan: You're very welcome. @Mathan: What kind of "curve" are you talking about? Ans so on. Its really a great job to post about quadratic equation and its curves..i ll recommend it to my colleagues. 1989) reserve the term for a quartic equation having no cubic term, i.e., a quadratic equation in x^2. if we fit these points generally they fit on parabola with axis of symmetry on Y axis but i want to fit these points in parabola with axis of symmetry on X Axis and 2 points of the parabola intersecting on Y Axis. Gimusi.Wonderful.I wish I could share anything with Lagrange. Free quadratic functions calculator. On parabola how can I find the equation of the axis of symmetry . I am confused about one thing.If the y-intercept is (4.2), would we replace the 4 in place if the x instead of zero.just making sure the 0 is not used every time. In the "Options" tab, choose "Display equation on chart". With a little perseverance, anyone can understand even the most complicated mathematical problems. Quadratic Equation in Standard Form: ax 2 + bx + c = 0 Quadratic Equations can be factored Quadratic Formula: x = b (b2 4ac) 2a When the Discriminant ( b24ac) is: positive, there are 2 real solutions zero, there is one real solution negative, there are 2 complex solutions Calculate a quadratic function given the vertex point. For example, 11 = (-b - c + 5)(2^2) + b(2) + c simplifies to b = -1.5c + 4.5. The possible x-values will be the x-intercepts; where you line crosses the x-axis. I was not aware of the FitPoly command in GeoGebra - it's a shame it is not included in one of the menus. Use the given point (-1, 3), which says y is 3 for x equal to -1. Also to see if you can use this to calculate sine values using two quadratic equations with one of them being the correction value add to the other to get it. Google Photos) then put the link to it here. It is an equation for the parabola shown higher up. Concluding this example, squaring (x - 2) results in. Substituting for x and y: 3 = a(-1 - 3)2 - 1 = 3 = a(-4)2 We find the vertex of a quadratic equation with the following steps: Get the equation in the form y = ax2 + bx + c. Calculate -b / 2a. If asked for the exact answer (as usually happens) and the square roots cant be easily simplified, keep the square roots in the answer, e.g. @billy: I added an extra line in there for you - hopefully it's clearer now. Given a quadratic equation, most algebra students could easily form a table of ordered pairs that describe the points on the parabola. Check out this video. What's the difference between a power rail and a signal line? @John: Yes, that would do it. Please let me know if this ok with you. a parabolic equation resembles a classic quadratic equation. Two equations are displayed: an exact one (top one) where the coefficients are in fractional forms an the second with approximated coefficients whose number of decimal number of decimal places may be chosen. Substituting a in the second equation yields 4 = 2b 2, which we simplify to b 2 = 2, or b = square root of 2, which equals approximately 1.41. Sincerely, Harry Dunleavy. 8 = a(3 - 2)^2 + 3 \text{ or } 8 = a(1)^2 + 3, University of Georgia: Writing Quadratic Equations. Hope it helps! For example, placing theentirenumerator over2aisnotoptional. Your work and problems are excellent. Keep up with the latest news and information by subscribing to our email list. Again, your page is interesting and excellent for H.S and college students. Figure 2: Quadratic formula. Calculate a quadratic function given the vertex point Computing a quadratic function out of three points. If all you knew was factoring, you would be stuck. First step, make sure the equation is in the format from above, The two solutions are the x-intercepts of the equation, i.e. Substituting for x and y: 3 = a(-1 - 3)2 - 1 = 3 = a(-4)2. Are there tables of wastage rates for different fruit and veg? How to Find The Quadratic Equation From a Table/Points Top Tier Math 796 subscribers Subscribe 117 Share 8.2K views 1 year ago Algebra 1 So, those fun problems where you're given a table. Writing Quadratic Equations for Given Points. Everything, from-bto the square root, is over2a. Or maybe it does have but the image provided is limited so I can't see it touching anything and I can't substitute. I have One question for the first method of systems of equations, where it says "Multiplying the last line by 2 and adding it to the line before gives in mathematics from the City University of New York and a masters from Long Island University. I am to find a equation of a parablo given the vertex (7,-2) and one x-intercept (4,0). Leave as is, rather than writing it as a decimal equivalent(3.16227766), for greater precision. In the standard form If we have a y-intercept, the we find it by substituting x = 0. (b) Use the graph to find the roots of the equation x-2x-3=0. I admire your desire to continue learning, however I don't think you'll find a reputable online PhD mathematics program. This set of data is a given set of graph points that make up the shape of a parabola. Method 1 Using the Vertex Formula 1 Identify the values of a, b, and c. In a quadratic equation, the term = a, the term = b, and the constant term (the term without a variable) = c. thank you. Just as a quadratic equation can map a parabola, the parabola's points can help write a corresponding quadratic equation. She has acted as a copywriter and screenplay consultant for Advent Film Group and as a promotional writer for Cinnamom Bakery. Math is a way of solving problems by using numbers and equations. Everytime i do this i get an infinite loop. I wish the camera feature was a bit more accurate, but other than that, it couldnt get any better, so it's not necessarily cheating if it helps you learn too, so far really good. Hi, I found your explanation lucid and helpful. Another approach to the parabola problem, which may be of particular interest to calculus students, is that for a parabola to be the graph of y=ax^2+bx+c: we are trying to find the equation of the parabola. For an example, let the vertex be (2, 3). More advanced: I want to fit parabola equation at any axis of symmetry. X^2+y^2+axy +bx + cy +k = 0 of the parabola on the graph, and plug it into the vertex form of a quadratic equation. The quadratic formula helps you solve quadratic equations, and is probably one of the top five formulas in math. Calculate a quadratic function given the vertex point Computing a quadratic function out of three points. Not only []. we are able to determine and establish goals. If you are going to try to do it algebraically for many different parabolas it's going to be quite troublesome. Therefore You would go about it in a similar way. http://www.sscc.edu/home/jdavidso/Math/Catalog/Polynomials/Fifth/FifthDegreeB.html, https://www.intmath.com/forum/plane-analytic-geometry-37/. So the y-intercept is 19. Ferrari was the first to develop an . If you need support, our team is available 24/7 . Our goal is to make science relevant and fun for everyone. Instead of x, you can also write x^2. 2 1 Sponsored by Forge of Empires For example, solving the equation for the points (0, 2) and (2, 4) yields: 2 = ab 0 and 4 = ab 2. 2021-11-06 Added 102 answers. Calculate a quadratic function given the vertex point Computing a quadratic function out of three points, Use the given point (-1, 3), which says y is 3 for x equal to -1. In this example, solving for a results in, Substitute the value of a into the equation from Step 1. Only the use of the In algebra, a quadratic equation is any polynomial equation of the second degree with the following form: ax 2 + bx + c = 0 where x is an unknown, a is referred to as the quadratic coefficient, b the linear coefficient, and c the constant. My math teacher said to solve for a as much as possible with one section, solve for b as much as possible in another, then uses them to solve eachother by plugging them in to eachother. How to find the equation of a quadratic function from its graph, New measure of obesity - body adiposity index (BAI), Whats the Best? [] Murray Bourne explains step by step How to find the equation of a quadratic function from its graph. Check. How do i know when the curve goes like a u or a upside down u ? The sum of the roots of a quadratic equation is + = -b/a. Learn more about Stack Overflow the company, and our products. NOTE: You can mix both types of math entry in your comment. Mathepower calculates the quadratic function whose graph goes through those points. 20+ tutors near you & online ready to help. It is written in the form: ax^2 + bx + c = 0 where x is the variable, and a, b, and c are constants, a 0. Direct link to Huron Tu's post In 1827, a mathematician , Posted a year ago. Posted in Mathematics category - 17 May 2011 [Permalink]. @Madhu: This is the same approach suggested by Paul, a few comments ago. We have (h, k) = (-2, 1) and at x = 0, y = 2. f(x) = 0.25(x (2))2 + 1 = 0.25(x + 2)2 + 1 = 0.25(x2 + 4x + 4) + 1. Good luck with your studies! Show more Show more Shop the Mario's Math Tutoring store Finding the. Thequadratic formulais an algebraic formula used to solve quadratic equations. In this article, we review how to graph quadratic functions. Direct link to Daniel Rendall's post does x2 = x to the power , Posted 9 years ago. Thanks! Sometimes it is easy to spot the points where the curve passes through, but often we need to estimate the points. . Assuming you're given three points along a parabola, you can find the quadratic equation that represents that parabola by creating a system of three equations. This means that at no point will. @Marisa: For your first question, this page will help: https://www.intmath.com/blog/mathematics/how-to-draw-y2-x-2-2301. We have to find the equation of the quadratic function that passes through the points ( 3, 0), ( 2, 0), and ( 0, 30) . The best thing is to supply the question and an example given by the teacher so I can see what they mean. For convenience let us assume that we have 3 points (1,5), (3,2) & (5,3). Hopefully this proof helps you understand why: There are several ways to derive the quadratic formula, but the simplest is by using completing the square. However, some may not realize you can also perform the reverse operation to derive the equation from the points. In this day of readily available (and free) computer tools, I no longer recommend Cramer's Rule! In this video the tutor shows how to find the mirror point using a quadratic equation. (a) Write down the turning point of the graph y=x - 2x - 3. What if your originalbisalreadynegative? Just curious, is there something like the "Trinomial formula", for third degree polynomials and so on? Very disappointing. All the best in your exam.