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Weekly online one to one GCSE maths revision lessons now available Learn more This topic is relevant for: Here we will learn about solving quadratic equations graphically including how to find the roots of a quadratic function from a graph, how to use this method to solve any quadratic equation by drawing a graph, and then how to solve a quadratic equation from a graph that is already drawn for you. There are also quadratic graphs worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck. What is solving quadratic equations graphically?Solving quadratic equations graphically is a useful way to find estimated solutions or roots for quadratic equations or functions. E.g. Solve x^2=6 graphically If we plot the quadratic function y=x^{2} and the linear function y=6 on the same graph, the intersection points of the line and the curve are the solutions to the quadratic equation x^{2}=6 . The solutions to the quadratic equation (to 1 dp) are x=-2.5, \; x=2.5 . You can see from the graph that these intersection points aren’t at exactly x=-2.5, \; x=2.5. These solutions are estimates. This is the main disadvantage of the graphical method – for exact solutions, we would need to use the quadratic formula or completing the square and leave answers in surd form. Step-by-step guide: Quadratic formula Step-by-step guide: Completing the square What is solving quadratic equations graphically?What are the roots of a quadratic?The roots of a quadratic function are the values of the x -coordinates where the function crosses the x -axis. They are sometimes called the x -intercepts. E.g. A quadratic function will have zero, one (repeated), or two real roots. When you solve a quadratic equation in the form ax^2+bx+c=0 , with the right-hand side equal to zero, you are finding the roots. On this page we will look at finding roots, and then finding other solutions, using a graph. How to solve quadratic equations graphicallyIn order to find the solutions of a quadratic equation using a graph:
Explain how to solve quadratic equations graphicallyQuadratic graphs worksheetGet your free quadratic graphs worksheet of 20+ questions and answers. Includes reasoning and applied questions. DOWNLOAD FREE Quadratic graphs worksheetGet your free quadratic graphs worksheet of 20+ questions and answers. Includes reasoning and applied questions. DOWNLOAD FREE Related lessons on quadratic graphsSolving quadratic equations graphically is part of our series of lessons to support revision on types of graphs. You may find it helpful to start with the main quadratic graphs lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:
Solve quadratic equations graphically examplesExample 1: a simple quadraticFind the solutions of the equation x^{2}-4=0 graphically
No rearrangement needed in this case. 2Draw the graph of the quadratic function. Write y=x^2-4 and draw the graph. 3Read off the x -coordinate(s) of the point(s) where the curve crosses the x -axis. The roots are x=-2 and x=2 . These are the solutions to the equation x^{2}-4=0 Example 2: a trinomial quadraticFind the solutions of the equation x^{2}+6x+9=0 graphically. Rearrange the equation so that one side = 0 (if necessary). No rearrangement needed in this case. Draw the graph of the quadratic function. Write y=x^{2}+6x+9 and draw the graph. Read off the x -coordinate(s) of the point(s) where the curve crosses the x -axis.
Example 3: no real rootsFind the solutions of the equation x^{2}-2x+4=0 graphically. Rearrange the equation so that one side = 0 (if necessary). No rearrangement needed in this case. Draw the graph of the quadratic function. Write y=x^{2}-2x+4 and draw the graph.
Read off the x -coordinate(s) of the point(s) where the curve crosses the x -axis. This quadratic function does not cross or touch the x -axis; there are no (real) roots and therefore the equation x^{2}-2x+4=0 has no (real) solutions.
Example 4: a rearrangementFind the solutions of the equation 12+x=x^{2} graphically. Rearrange the equation so that one side = 0 (if necessary). Bring the x^2 to the LHS to give 12+x-x^{2}=0 Draw the graph of the quadratic function. Write y=12+x-x^{2} and draw the graph. Read off the x -coordinate(s) of the point(s) where the curve crosses the x -axis.
Example 5: a rearrangementFind the solutions (to 1 decimal place) of the equation 3-2x^{2}=4x graphically. Rearrange the equation so that one side = 0 (if necessary). Bring the 4x to the LHS to give 3-2x^{2}-4x=0 Draw the graph of the quadratic function Write y=3-2x^{2}-4x and draw the graph.
Read off the x -coordinate(s) of the point(s) where the curve crosses the x -axis.
Solving a quadratic equation when a graph is givenThe method detailed above will always work, for any quadratic equation – you can rearrange so that one side equals 0 , plot the points and find the roots. However, sometimes you may already have drawn a particular graph or had this given to you – this is usually the case in exam questions. In these situations, this alternative method is quicker. In order to find solutions to a quadratic equation using a graph:
Example 6: a quadratic and a horizontal lineHere is the graph of the function y=x^{2} . Use this graph to find the solutions of the equation x^{2}=3 . Rearrange so that one side of the equation matches the graphed function. The equation is x^{2}=3 and the graph we’re given is of the function y=x^{2} . The LHS of the original equation already matches the RHS of the function on the graph. In this case, we don’t need to rearrange. Write y = the other side of the equation and plot this function. Write y=3 and plot this.
At the intersection points, draw vertical lines down to the x -axis to find the solutions. Find the two points where the line and curve meet; we draw vertical lines down to the x -axis and read off the x -coordinate values.
Example 7: a quadratic and line with non-zero gradientHere is the graph of the function y=x^{2}-2x+4 . (This is the same graph we used in Example 3). Find the solutions of the equation x^{2}-2x+4=2x+4 graphically. Rearrange so that one side of the equation matches the graphed function. The equation is x^{2}-2x+4=2x+4 and the graph we’re given is of the function y=x^{2}-2x+4 . The LHS of the original equation already matches the RHS of the function on the graph. In this case, we don’t need to rearrange. Write y = the other side of the equation and plot this function. Write y=2x+4 and plot this.
y = mx + c
At the intersection points, draw vertical lines down to the x -axis to find the solutions. Find the two points where the line and curve meet; we draw vertical lines down to the x -axis and read off the x -coordinate values.
Example 8: rearrangement requiredThis is an example of the sort of question favoured in exams; the graph of one function is pre-drawn, and you will have to manipulate the algebra in some way in order to be able to use the graph. Here is the graph of the function y=x^{2}+3x+2 . Find the solutions of the equation x^{2}+3x=-1 graphically, to 1 decimal place. Rearrange so that one side of the equation matches the graphed function. The equation is x^{2}+3x=-1 and the graph we’re given is of the function y=x^{2}+3x+2 .
Write y = the other side of the equation and plot this function. Write y=1 and plot this.
At the intersection points, draw vertical lines down to the x -axis to find the solutions. Find the two points where the line and curve meet; we draw vertical lines down to the x -axis and read off the x -coordinate values.
Common misconceptions
Make sure that the vertex of the graph is a smooth curve, not pointed:
In order to solve when you haven’t been given a graph, rearrange so that one size equals 0 , then find the roots. In order to solve when you have been given a graph, rearrange so that one side of the equation matches the function that’s been graphed. Practice solving quadratic equations graphically questionsThe RHS is already 0 , so plot the function on a graph and find the roots. The RHS is already 0 , so plot the function on a graph and find the roots. The RHS is already 0 , so plot the function on a graph and find the roots. These solutions are estimates, so give them carefully to 1 decimal place.
Rearrange so that RHS = 0 , and plot the graph of y=x^{2}-2x-5 . Give solutions to 1 decimal place. Rearrange the equation to get x^{2}+4x-3=1 , then draw the line y=1 on the graph given. Read off solutions to 1 decimal place. Rearrange the equation to get 7x-2x^{2}=0 . Because RHS = 0 , you can just read off the roots on the x -axis. Rearrange the equation to get x^{2}+x-3=2 x+7 . Read off the x values where the line and curve intersect. Solving quadratic equations graphically GCSE questions1. Here is the graph of y=6x-x^{2} (a) Use the graph to write down the roots of the equation 6x-x^{2}=0 (b) Use the graph to find the solutions of the equation 6x-x^{2}=3 , correct to 1 decimal place. (5 marks) Show answer (a) Indication of reading x -intercept values/roots (1) x=0, \; x=6 (1) (b) Line y=3 drawn on graph (1) Indication of reading coordinates of intersection points (1) x=0.6, \; x=5.5 (1) 2. Here is the graph of y=x^{2}-4x+2 Use the graph to find approximate roots of the equation x^{2}+2=4x (2 marks) Show answer Indication of reading x -intercept values (1) x=0.6,\; x=3.4 (1) 3. (a) Complete the table of values for y=x^{2}+2x-6 \begin{aligned} &x \quad -4 \quad -3 \quad -2 \quad -1 \quad \quad 0 \quad \quad 1 \quad \quad 2 \\ &y \quad \quad \quad \quad \quad \quad -6 \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\; 2 \end{aligned} (b) On the grid draw the graph of y=x^{2}+2x-6 for values of x from -4 to 2 . (c) Use the graph to find estimates of the solutions to the equation -2=x^{2}+2x-6 . (d) Use the graph to find solutions to the equation (9 marks) Show answer (a) \begin{aligned} &x \quad -4 \quad -3 \quad -2 \quad -1 \quad \quad 0 \quad \quad 1 \quad \quad 2 \\ &y \quad \quad 2 \quad -3 \quad -6 \quad -7 \quad -6 \quad -3 \quad \quad 2 \end{aligned} (2) (b) Points plotted correctly ft. pt (a) (1) Points joined with a smooth curve (1) (c) Line y=-2 drawn on graph (1) x=-3.2, \; x=1.2 (1) (d) Rearrange x^{2}+2 x=3 to x^{2}+2 x-6=-3 (1) Line y=-3 drawn on graph (1) x=-3, \; x=1 (1) Learning checklistYou have now learned how to:
Still stuck?Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors. Find out more about our GCSE maths revision programme. We use essential and non-essential cookies to improve the experience on our website. Please read our Cookies Policy for information on how we use cookies and how to manage or change your cookie settings.Accept How do you find the solutions of a quadratic?Solving Quadratic Equations. Put all terms on one side of the equal sign, leaving zero on the other side.. Factor.. Set each factor equal to zero.. Solve each of these equations.. Check by inserting your answer in the original equation.. |