Topics
8. Solving Quadratic Equations
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United States of AmericaTennessee
Algebra 1

8.01 Solving quadratic equations using graphs and tables

Lesson
Practice
Lesson

Concept summary

A quadratic equation is a polynomial equation of degree 2. The standard form of a quadratic equation is written in the in the form ax^2+bx+c=0 where a, b, and c are real numbers.

We can solve some quadratic equations by drawing the graph of the corresponding function. This also allows us to determine the number of real solutions it has.

x
y
One real solution
x
y
Two real solutions
x
y
No real solutions

The solutions to a quadaratic equation are the x-intercepts of the corresponding function. They also known as the roots of the equation or the zeros of the function. A quadratic equation with no real solutions is said to have non-real solutions.

The zeros of an equation can be also be seen in a table of values, provided the right values of x are chosen, and the equation has at least one real solution.

Worked examples

Example 1

Complete a table of values for y=2x^2-18 and then determine the solutions to the corresponding equation 2x^2-18=0.

Approach

When building a table we want to choose values with 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 function value is zero for any of these x-values, then we have found a solution to the corresponding equation.

Solution

x-4-3-2-101234
y140-8-16-18-16-8014

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

Reflection

We can see that the table of 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.

Approach

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.

Solution

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

Reflection

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.

Approach

The solutions to the equation 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.

Solution

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

Reflection

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.

Approach

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).

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

Solution

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.

Reflection

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.

Outcomes

A1.N.Q.A.1

Use units as a way to understand real-world problems.*

A1.N.Q.A.1.A

Choose and interpret the scale and the origin in graphs and data displays.*

A1.A.CED.A.1

Create equations and inequalities in one variable and use them to solve problems in a real-world context.*

A1.A.CED.A.2

Create equations in two variables to represent relationships between quantities and use them to solve problems in a real-world context. Graph equations with two variables on coordinate axes with labels and scales, and use the graphs to make predictions.*

A1.A.CED.A.3

Create individual and systems of equations and/or inequalities to represent constraints in a contextual situation, and interpret solutions as viable or non-viable.*

A1.A.REI.B.3

Solve quadratic equations and inequalities in one variable.

A1.A.REI.B.3.A

Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, knowing and applying the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when a quadratic equation has solutions that are not real numbers.

A1.A.REI.D.5

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A1.A.REI.D.6

Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x). Find approximate solutions by graphing the functions or making a table of values, using technology when appropriate.*

A1.MP2

Reason abstractly and quantitatively.

A1.MP3

Construct viable arguments and critique the reasoning of others.

A1.MP4

Model with mathematics.

A1.MP5

Use appropriate tools strategically.

A1.MP6

Attend to precision.

A1.MP7

Look for and make use of structure.

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