solving quadratic equations by graphing practice

Solving Quadratic Equations by Graphing - Examples & Practice - Expii

Solving quadratic equations by graphing - examples & practice, explanations (3).

Solving quadratics by graphing is a great complement other approaches, like factoring . You can use it to verify that your solutions are correct.

Regardless of the approach you take, "solving quadratic equations" means solving for possible values of x.

Recall that the graph of a quadratic equation is a parabola (U shapes). The solutions to the equation (i.e., the possible values of x) can be found by locating the x-intercepts of the parabola. (Note that we are looking at x-intercepts here, not y-intercepts.)

Let's take a look at y=x2+4x−5:

Image source: Desmos

We need to locate the parabola's x-intercepts. These are our solutions.

The x-intercepts are located where the parabola crosses the x-axis.

In our graph, the parabola crosses the x-axis at x=−5 and x=1.

So, the solution to the quadratic equation is: x=−5,1

Related Lessons

(Video) Solving Quadratic Equations by Graphing

by Textbook Tactics

This video by Textbook Tactics shows you how to solve a quadratic by graphing.

We normally see quadratics written as: y=ax2+bx+c When we want to solve them we set them equal to 0. ax2+bx+c=0 This means that the solutions to the quadratic are any point where: y=0

We can graph the parabola and find the the points where it crosses the x-axis. These points are called the x-intercepts, which you may remember from linear graphs .

The first example problem is

We need to graph this equation, so lets start with finding the vertex. The x coordinate is found with x=−b2a. The x-value also gives us our axis of symmetry . Let's plug in our values.

x=−b2ax=−(−2)2(1)x=22x=1

The x-coordinate of the vertex is 1. The axis of symmetry is x=1. Let's plug 1 into the equation to find the y-coordinate of the vertex.

y=x2−2x−3y=(1)2−2(1)−3y=1−2−3y=−4

The vertex is at (1,−4). Let's put these on the graph.

We need to find another point to graph this. Let's find the y-intercept by setting x=0.

y=x2−2x−3y=(0)2−2(0)−3y=−3

The y-intercept is (0,−3). This is one unit to the left of the vertex, so we know that the point one unit to the right of the vertex also has a y-value of −3. That point will be (2,−3). Let's plot these.

We can keep extending the plot of this graph by plugging in more x-values. Let's plug in x=3 and x=−1, since they are 2 unites to the right and left of the vertex, respectively.

y=x2−2x−3y=(3)2−2(3)−3y=9−6−3y=0y=x2−2x−3y=(−1)2−2(−1)−3y=1+2−3y=0

We can add the points (3,0) and (−1,0) to our graph, along with the parabola that goes through all of these points.

The last two points are where the parabola crosses the x-axis. That means the solutions to y=x2−3x−3 are −1 and 3.

Remember, not all quadratics have 2 solutions. A quadratic with 1 solution will touch the x-axis in exactly one point. A quadratic with no real solutions will never cross the x-axis. The video linked above has more examples including quadratics with 1 and no solutions.

Solving Quadratic Equations by Graphing

All quadratic equations have at most two solutions — these solutions can be real or imaginary . Some quadratic equations have two real solutions, some have two imaginary solutions, and some have one real solution, often called a double root or a repeated root. (It is never the case that a quadratic equation can have one real solution and one imaginary solution.)

There are multiple methods to find the solutions to a quadratic equation, including factoring , using the quadratic formula , completing the square , and graphing.

Solutions = x-Intercepts

Say you have access to a graphing calculator, and you want to quickly find the solutions to a quadratic equation. Luckily, when you graph a quadratic equation, it's easy to see what its real solutions are! All you have to do is look at the x-intercepts...those are the solutions! (A reminder: the x-intercepts are the spots on the graph where the parabola intersects with the x-axis.)

One x-intercept means the quadratic equation has one real solution, and that solution is the value of that x-intercept.

Two x-intercepts means that the quadratic equation has two real solutions, and those solutions are the values of those x-intercepts.

No x-intercepts means that the quadratic equation has no real solutions...it must have two imaginary solutions!

Image source: By Caroline Kulczycky

Let's do an example! Say you are given the quadratic equation y=x2−x−6, and you want to know its solutions. It's a bit difficult to simply look at this equation and immediately see the solutions, so let's graph it!

Here, we can see that there are two x-intercepts: x=−2 and x=3. Those are the solutions to y=x2−x−6!

You can use the graphing technique to solve linear equations, cubic equations, and even systems of equations (like linear-quadratic systems )!

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Solving Quadratic Equations Graphically

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.

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 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.

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 graphically

In order to find the solutions of a quadratic equation using a graph:

Rearrange the equation so that one side = 0 (if necessary).

• Draw the graph of the quadratic function.

Read off the x -coordinate(s) of the point(s) where the curve crosses the x -axis.

Explain how to solve quadratic equations graphically

Quadratic graphs worksheet

Get your free quadratic graphs worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on quadratic graphs

Solving 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:

• Types of graphs
• Quadratic graphs
• Sketching quadratic graphs
• Plotting quadratic graphs

Solve quadratic equations graphically examples

Example 1: a simple quadratic.

Find the solutions of the equation x^{2}-4=0 graphically

Rearrange the equation so that one side = 0 (if necessary) .

No rearrangement needed in this case.

2 Draw the graph of the quadratic function .

Write y=x^2-4 and draw the graph.

3 Read 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 quadratic

Find the solutions of the equation x^{2}+6x+9=0 graphically.

Draw the graph of the quadratic function .

Write y=x^{2}+6x+9 and draw the graph.

Here we have one (repeated) root at x=-3 .

The solution to the equation x^{2}+6 x+9=0 is x=-3 .

Example 3: no real roots

Find the solutions of the equation x^{2}-2x+4=0 graphically.

Write y=x^{2}-2x+4 and draw the graph.

Step by step guide : Plotting quadratic graphs (Example 5)

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.

If you tried to solve the equation x^{2}-2x+4=0 using the quadratic formula, this would result in a ‘maths error’ when entered into your calculator.

You will see “no real roots” written because these equations can be solved using techniques studied in A Level Further Mathematics, so it’s not correct just to write “there are no roots”. 

Example 4: a rearrangement

Find the solutions of the equation 12+x=x^{2} graphically.

Bring the x^2 to the LHS to give 12+x-x^{2}=0

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 .

The roots are x=-3 and x=4 . .

These are the solutions to the equation 12+x=x^{2} .

Example 5: a rearrangement

Find the solutions (to 1 decimal place) of the equation 3-2x^{2}=4x graphically.

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.

Step by step guide: Plotting quadratic graphs (Example 7)

Here, the curve doesn’t cross at integer or easy-to-read values. Give the roots as estimates; you shouldn’t try to be any more accurate than one decimal place.

The roots are x=-2.6 and x=0.6 . 

These are approximate solutions to the equation 3-2x^{2}=4x , correct to one decimal place.

Solving a quadratic equation when a graph is given

The 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:

• Rearrange so that one side of the equation matches the graphed function.
• Write y = the other side of the equation and plot this function.

At the intersection points, draw vertical lines down to the x -axis to find the solutions.

Example 6: a quadratic and a horizontal line

Here 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.

It is a horizontal line through the y -axis at 3 :

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.

These values aren’t particularly easy to read off. Give the roots as estimates; you shouldn’t try to be any more accurate than one decimal place.

We have solutions x=-1.7, \; x=1.7 , correct to 1 decimal place.

Example 7: a quadratic and line with non-zero gradient

Here 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.

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=2x+4 and plot this.

It is a straight line with y -intercept (0,4) and gradient 2 .

Step by step guide: Plotting linear graphs (coming soon)

We have solutions x=0, \; x=4 .

Example 8: rearrangement required

This 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.

The equation is x^{2}+3x=-1 and the graph we’re given is of the function y=x^{2}+3x+2 .

These look similar; there’s an x^2 and a +3x term in both, but we need to rearrange so that the constant term is also the same, and the LHS of the equation matches the RHS of the function.

We need a +2 on the LHS of the equation, so we add 2 to both sides of the equation:

Write y=1 and plot this.

It is a horizontal line through the y -axis at 1 :

We have (estimated) solutions x=-2.6, \; x=-0.4 .

These are the solutions to the original equation, x^{2}+3x=-1 , correct to 1 decimal place.

Common misconceptions

• Drawing a pointy vertex

Make sure that the vertex of the graph is a smooth curve, not pointed:

• Forgetting to rearrange when necessary

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 questions

1. Find the solutions of the equation x^{2}-9=0 graphically.

The RHS is already 0 , so plot the function on a graph and find the roots.

2. Find the solutions of the equation x^{2}+5x+4=0 graphically.

3. Find the solutions of the equation 3 x-x^{2}+2=0 graphically to 1 decimal place.

These solutions are estimates, so give them carefully to 1 decimal place.

4. Find the solutions of the equation x^{2}-2x=5 graphically.

Rearrange so that RHS = 0 , and plot the graph of y=x^{2}-2x-5 . Give solutions to 1 decimal place.

5. Here is the graph of y=x^{2}+4 x-3 .

Use the graph to find the solutions of the equation x^{2}+4x=4 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.

6. Here is the graph of y=7x-2x^{2} .

Use the graph to find the solutions of the equation 7x=2x^{2} 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.

7. Here is the graph of y=2x^{2}+x-3 .

Use the graph to find the solutions of the equation 2x^{2}-x=10 to 1 decimal place.

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 questions

1. 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.

Indication of reading x -intercept values/roots

x=0, \; x=6

Line y=3 drawn on graph

Indication of reading coordinates of intersection points

x=0.6, \; x=5.5

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

Indication of reading x -intercept values

x=0.6,\; x=3.4

3.    (a) Complete the table of values for y=x^{2}+2x-6

(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 x^{2}+2x=3 .

Points plotted correctly ft. pt (a)

Points joined with a smooth curve

Line y=-2 drawn on graph

x=-3.2, \; x=1.2

Rearrange x^{2}+2 x=3 to x^{2}+2 x-6=-3

Line y=-3 drawn on graph

x=-3, \; x=1

Learning checklist

You have now learned how to:

• Find the roots of a quadratic function graphically
• Find approximate solutions to quadratic equations using a graph

The next lessons are:

• Factorising
• Solving equations

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Course: Algebra 1   >   Unit 14

• Forms & features of quadratic functions
• Worked examples: Forms & features of quadratic functions
• Features of quadratic functions: strategy
• Vertex & axis of symmetry of a parabola
• Finding features of quadratic functions
• Features of quadratic functions

Graph parabolas in all forms

• Interpret quadratic models: Factored form
• Interpret quadratic models: Vertex form
• Interpret quadratic models
• Graphing quadratics review
• Creativity break: How does creativity play a role in your everyday life?

• List of Lessons
• 1.0 Marking the Text in Mathematics
• 1.1 Order of Operations
• 1.2 Translating Verbal Phrases
• 1.3 Functions as Rules and Tables
• 1.4 Functions as Graphs
• Unit 1 Review
• Unit 1 Skillz Review Help
• 2.1 Real Numbers
• 2.2 Add and Subtract Real Numbers
• 2.3 Multiply and Divide Real Numbers
• 2.4 Combine Like Terms and Distributive Property
• Unit 2 Review
• Unit 2 Skillz Review
• 3.1 One Step Equations
• 3.2 Two Step Equations
• 3.3 MultiStep Equations
• 3.4 Variables on Both Sides of the Equation
• Unit 3 Review
• 4.1 Ratios and Proportions
• 4.2 Solving Proportions using Cross Products
• 4.3 Solving Percent Problems
• 4.4 Solving for Y
• Unit 4 Shortcuts
• Unit 4 Review
• 5.1 Plot Points in a Coordinate Plane
• 5.2 Graphing Linear Equations and Using Intercepts
• 5.3 Slope and Rate of Change
• 5.4 Graph in Slope-Intercept Form
• 5.5 Graph Linear Functions
• Unit 5 Review
• 6.1 Write Equations in Slope Intercept Form
• 6.2 Use Equations in Slope Intercept Form
• 6.3 Equations in Parallel/Perpendicular Form
• 6.4 Fit Line To Data
• Unit 6 Review
• Unit 6 Skillz Review Help
• SEMESTER EXAM REVIEW
• 7.1 Inequalities
• 7.2 Solving Inequalities
• 7.3 Multi-step Inequalities
• 7.4 Absolute Value Equations
• 7.5 Linear Inequalities in Two Variables
• Unit 7 Skillz Review Help
• 8.1 Solving Systems by Graphing
• 8.2 Solving Systems using Substitution
• 8.3 Solving Systems using Elimination
• 8.4 Solving Special Case Systems
• 8.5 Solving Systems of Linear Inequalities
• Unit 8 Review
• 9.1 Expand and Condense Exponents
• 9.2 Exponent Rules
• 9.3 Zero and Negative Exponents
• 9.4 Scientific Notation
• Unit 9 Review
• 10.1 Add and Subtract Polynomials
• 10.2 Multiply Polynomials
• 10.3 Polynomial Equations in Factored Form
• 10.4 Factoring Trinomials
• 10.5 Solving Quadratics by Factoring
• 10.6 Double Factoring
• Unit 10 Review
• Unit 10 Skillz Review Help
• 11.1 Simplifying Radicals
• 11.2 Operations with Square Roots
• Unit 11 Review
• 12.1 Graphing Quadratics

12.2 Solve Quadratics by Graphing

• 12.3 Solve Quadratics Using Square Roots
• 12.4 Solve Quadratics Using the Quadratic Formula
• Unit 12 Review
• Unit 12 Skillz Review
• SEMESTER 2 EXAM REVIEW
• Teacher Resources
• Flippedmath.com

Practice Solutions:

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