Topics
11. Quadratic Equations
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United States of America
Grades 9-12

11.01 Solving using graphs and tables

Lesson
Practice

Introduction

We previously explored the key features of quadratic functions in various forms. Each of the forms highlighted important features of the graph. Now, we will learn how to solve quadratic equations using various strategies. Regardless of which form the quadratic equation is in, we can use a graph or a table to solve the equation.

Solving quadratic equations using graphs and tables

A quadratic equation is a polynomial equation of degree 2. The solutions to a quadratic equation are the values that make the equation true.

A table with 3 columns titled x, x squared equals y, and Point, and with 7 rows. The data is as follows: First row: negative 2.8, quantity negative 2.8 squared equals 7.84, (negative 2.8, 7.84); Second row: negative 2, quantity negative 2 squared equals 4, (negative 2, 4); Third row: negative 3 halves, quantity negative 3 halves squared equals 9 over 4, (negative 3 halves, 9 over 4); Fourth row: 0, 0 squared equals 0, (0, 0); Fifth row: 2 thirds, 2 thirds squared equals 4 over 9, (2 thirds, 9 over 4); Sixth row: 2.4, 2.4 squared equals 5.76, (2.4, 5.76); Seventh row: 3, 3 squared equals 9, (3, 9).
Some solutions to y=x^2 are shown in the table
The graph of y equals x squared plotted on a first and second quadrant coordinate plane. The graph is shown passing through the following points: (-2.8, 7.84), (-2, 4), (-1.5, 2.25), (0, 0), (0.67, 0.44), (2.4, 5.76), and (3, 9).
Solutions to y=x^2 are any point on the curve

The solutions to a quadratic equation where y is equal to zero are the x-intercepts of the corresponding function. They are also known as the roots of the equation or the zeros of the function.

A table with 2 columns titled x and y, and with 7 rows. The data is as follows: First row: negative 3, 9; Second row: negative 2, 4; Third row: negative 1, 1; Fourth row: 0, 0; Fifth row: 1, 1; Sixth row: 2, 4; Seventh row: 3, 9. The fourth row containing 0, 0 is labeled Solution to x squared equals 0.
Table of y=x^2
The graph of y equals x squared plotted on a first and second quadrant coordinate plane. The point (0, 0) is labeled Solution to x squared equals 0.
Graph of y=x^2

In the graph and table above, we see x=0 is the only solution. This is because x=0 is the only value that makes x^2=0 true.

If we tried to find the solution to x^2=-2 there would be no real solutions, because squaring any non-zero real number will give a positive result.

Often, there are two solutions to equations involving quadratics. If we want to find the solution to the equation x^2=4, we look for the x-values that make the y-values equal to 4. This time, there are two answers: x=-2 and x=2. Substituting them into the equation, we can see that both answers make the equation true: (-2)^2=4 and (2)^2=4.

A table with 2 columns titled x and y, and with 7 rows. The data is as follows: First row: negative 3, 9; Second row: negative 2, 4; Third row: negative 1, 1; Fourth row: 0, 0; Fifth row: 1, 1; Sixth row: 2, 4; Seventh row: 3, 9. The second row containing negative 2, 4 and the sixth row containing 2, 4 are both labeled Solution to x squared equals 4.
Table of y=x^2
The graph of y equals x squared plotted on a first and second quadrant coordinate plane. The points (negative 2, 4) and (2,4) are labeled Solution to x squared equals 4.
Graph of y=x^2

Another method to solve x^2=4 is by creating an equivalent equation and then identifying the corresponding transformation in the graphs. If we set this equation equal to zero, we would get

\displaystyle x^2\displaystyle =\displaystyle 4Given equation
\displaystyle x^2-4\displaystyle =\displaystyle 0Subtraction property of equality

By setting the equation equal to zero, we can now consider the graph of y=x^2-4. This equation is the graph of y=x^2 shifted down 4 units. Shifting a graph vertically does not change the x-values of the function. Therefore, if we find the x-intercepts of y=x^2-4, then we will also find the x-coordinates that solve x^2 = 4.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y

For any function f(x)=c, for some real number constant, c, we can write the equivalent equation f(x)-c=0, and find the x-intercepts of g(x) = f(x) - c to solve for x.

Examples

Example 1

Solve the equation 2x^2 = 18.

Worked Solution
Create a strategy

We can write an equivalent equation set equal to zero, and then use a table to find the zeros of the new function.

Apply the idea

If we set this equation equal to zero, we would get

\displaystyle 2x^2\displaystyle =\displaystyle 18Given equation
\displaystyle 2x^2-18\displaystyle =\displaystyle 0Subtraction property of equality

When building a table, we want to choose values within a suitable range so we don't have to do too many calculations. Start by finding the values in the domain -4\leq x\leq4. If the y-value (also called the function value) is zero for any of these x-values, then we have found a solution to the corresponding equation.

In the table, we are looking for the entries where y=0.

x-4-3-2-101234
y140-10-16-18-16-10014

We can see the equation has solutions of x=-3,x=3, which we can also write as x= \pm 3.

Reflect and check

We can see that the table of y-values has both positive and negative values. Whenever this is the case for a function of the form f\left(x\right)=ax^2+bx+c, we know that the equation 0=ax^2+bx+c must have two real solutions, and the corresponding parabola will have two x-intercepts.

Example 2

Consider the function y=\left(x-2\right)^2-9.

a

Draw a graph of the function.

Worked Solution
Create a strategy

The function is given in vertex form so we know the vertex is at \left(2, -9\right). We can substitute x=0 to find the y-intercept at \left(0, -5\right). We can find other points on the curve by substituting in other values, and by filling a table of values.

Apply the idea
x01234
y-5-8-9-8-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
-10
-8
-6
-4
-2
2
4
6
y
Reflect and check

It is important when drawing graphs to clearly show the key features such as the vertex and the intercepts by choosing appropriate scales for the axes.

b

Determine the solution to the equation \left(x-2\right)^2=9.

Worked Solution
Create a strategy

One way to solve this equation is by creating an equivalent equation. We can subtract 9 from both sides to get it to be equal to zero. This would give us \left(x-2\right)^2-9=0 which corresponds to the function we graphed in part (a).

Apply the idea

The solutions to the equation (x-2)^2=9 can be found at the x-intercepts of the graph we just drew, as \left(x-2\right)^2-9=0 is an equivalent equation.

-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
-10
-8
-6
-4
-2
2
4
6
y

The solutions are x=-1 and x=5.

Reflect and check

We will verify our solutions by comparing the table of values for the function in part (a) and the function y=(x-2)^2.

x-1012345
y=(x-2)^29410149
y=(x-2)^2-90-5-8-9-8-50

(x-2)^2=9 at x=-1 and x=5

(x-2)^2-9=0 at x=-1 and x=5

In this case, the equation has integer solutions, which makes solving graphically an effective method. In other cases, the answer can be irrational and so solving graphically may only help determine an approximate solution.

c

Using the previous parts, predict whether the equation \left(x-2\right)^2-9=4 has real solutions. If it does, determine how many solutions it will have.

Worked Solution
Create a strategy

In part (b), we noticed that \left(x-2\right)^2-9=0 has solutions which correspond to the x-intercepts of the graph drawn in part (a).

Remember that x-intercepts are the points on the graph where y=0.

Apply the idea

The graph in part (a) is for y=\left(x-2\right)^2-9, so the solutions to \left(x-2\right)^2-9=4 will be the points where y=4.

-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
x
-10
-8
-6
-4
-2
2
4
6
y

We can see that the line y=4 intersects the parabola at two points, so we can predict that the equation \left(x-2\right)^2-9=4 will have two real solutions, one for each unique point of intersection.

Reflect and check

Notice that the function y=\left(x-2\right)^2-9 has a range of y\geq -9.

Since y=4 is in the range, we can predict that \left(x-2\right)^2-9=4 will have real solutions.

Example 3

The graph below shows the path of a rock after it has been thrown from a cliff where x represents the time in seconds and y represents the height of the rock in feet.

Height of the rock
2
4
6
8
10
\text{Time (}x\text{, seconds)}
4
8
12
16
20
24
28
32
36
40
\text{Height (}y\text{, ft)}

Determine when the rock has a height of 38 feet.

Worked Solution
Create a strategy

We are looking for when the rock reaches a height of 38 feet. In other words, we are given the height (the y-value), and we are looking for the time (the x-values).

Apply the idea

Notice that there are 2 places where the graph has a y-value of 38. The solutions will be the x-values of those 2 points.

Height of the rock
2
4
6
8
10
\text{Time (}x\text{, seconds)}
4
8
12
16
20
24
28
32
36
40
\text{Height (}y\text{, ft)}

The rock reaches a height of 38 feet after 2 seconds and 6 seconds.

Reflect and check

We can use the vertex to determine the equation of this graph. The vertex is at (4, 42). The parabola is facing downard, so we know a will be negative. We can use the vertex form to write an equation:

y=a(x-4)^2+42

We can plug in the coordinates for another known point, like the y-intercept, which is (0, 26) and then solve for a. In this case, the result will be a = -1. We just found the solutions to the corresponding equation -(x-4)^2+42=38.

Idea summary

The solutions to a quadratic equation are any values that make the equation true.

When using a table or a graph to solve a quadratic equation that is equal to a number, we look for where the y-values are equal to that number. The solutions are the associated x-values.

If the equation is equal to 0, the solutions are called roots of the equation or zeros of the function. These correspond to the x-intercepts of the graph.

For any function f(x)=c, for some real number constant, c, we can write the equivalent equation f(x)-c=0, and find the x-intercepts of g(x) = f(x) - c to solve for x.

Outcomes

A.CED.A.1

Create equations and inequalities in one variable and use them to solve problems.

A.CED.A.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A.REI.B.4

Solve quadratic equations in one variable.

F.IF.C.7.A

Graph linear and quadratic functions and show intercepts, maxima, and minima.

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