Lesson 11 All Things Being Equal Develop Understanding

Learning Focus

Solve quadratic equations graphically and algebraically.

Make connections between solving quadratic equations and graphing quadratic functions.

How can we use graphs to solve quadratic equations?

Open Up the Math: Launch, Explore, Discuss

As we have learned, quadratic functions can be very useful models for a lot of real situations. They are part of understanding the motion of objects, business and economic models, and many other applications. Using quadratic functions often requires solving a quadratic equation. This can be a pretty straightforward process, or it can be a little complicated. The good news is that we know a lot about quadratic functions that we can apply to solving equations. Let’s get started.

1.

If you were given the quadratic equation, $x^{2} = 9$ , how would you solve it? What solution(s) would you get?

2.

Graph the function: $f (x) = x^{2}$ .

a.

b.

For what values of $x$ does $f (x) = 9$ ? How did you find the values? How do they compare to the solutions of the equation?

3.

Given the equation ${(x - 1)}^{2} = 4$ :

a.

Solve the equation algebraically.

b.

Graph the function $f (x) = {(x - 1)}^{2}$ and mark the points where $f (x) = 4$ .

4.

Given the equation: $- x^{2} - 6 x - 7 = 1$

a.

Graph the function $f (x) = - x^{2} - 6 x - 7$ and find where $f (x) = 1$ . (Just to be efficient, use technology and record your graph here with the solution points marked.)

b.

Solve the equation algebraically.

Don’t feel bad if you couldn’t solve the equation algebraically. It’s one of those complicated equations that can’t be solved directly with inverse operations. Knowing that the graph of an equation is the set of all of its solutions, we can solve any quadratic equation graphically using technology. Now we’re going to use our understanding of the graph of a quadratic function and its symmetry to develop other algebraic techniques for solving quadratic equations.

Let’s start in a familiar place.

5.

If you’re given the function $f (x) = (x - 4) (x + 3)$ and asked to find the $x$ -intercepts, what quadratic equation would you be solving?

6.

How could you use this kind of thinking to solve: $x^{2} - x - 12 = 0$ ?

7.

Now you can try a few. Solve each quadratic equation by factoring.

a.

$x^{2} - 9 x + 20 = 0$

b.

$3 x^{2} - 14 x - 5 = 0$

c.

$2 x^{2} + 8 x - 42 = 0$

When we were graphing, we saw that some functions are easier to factor, and others are easier to complete the square. Let’s start with one where the work has been done for us so that we can start seeing some relationships.

8.

Consider the function: $f (x) = {(x - 1)}^{2} - 4$ and the equation $0 = {(x - 1)}^{2} - 4$ .

a.

Graph the function:

b.

Line of symmetry:

c.

Vertex:

d.

What are the $x$ -intercepts of $f (x)$ ?

e.

How far are the $x$ -intercepts from the line of symmetry? (Think of the distance left and right.)

f.

Use inverse operations to solve the equation $0 = {(x - 1)}^{2} - 4$ .

g.

How does solving the equation relate to the graph and the $x$ -intercepts?

Don’t be mad, but that equation would have factored easily if it were given in standard form. That’s why the solutions were integers. Not all equations factor easily, and the relationship that we are beginning to see can be very helpful.

Let’s look at a function and equation that do not factor easily from standard form, and see how vertex form can help us solve the equation or find the $x$ -intercepts.

9.

Start with the function $f (x) = x^{2} - 6 x + 4$ .

a.

Graph the function by putting it in vertex form.

b.

Vertex:

c.

Line of symmetry:

d.

What do you estimate the $x$ -intercepts to be?

e.

What do you estimate the distance from each $x$ -intercept to the line of symmetry to be?

f.

Starting with $f (x)$ in vertex form, write and solve the equation $f (x) = 0$ using inverse operations.

g.

How do your solutions compare with your estimates of the $x$ -intercepts and the distance from the line of symmetry and the $x$ -intercepts?

10.

Given: $x^{2} - 6 x + 7 = 0$

a.

Solve the equation by completing the square and using inverse operations.

b.

Let $f (x) = x^{2} - 6 x + 7$ . What is the line of symmetry?

c.

What is the distance from line of symmetry to an $x$ -intercept?

11.

Given: $2 x^{2} - 8 x + 5 = 0$

a.

Solve the equation by completing the square and using inverse operations.

b.

Let $f (x) = 2 x^{2} - 8 x + 5$ . What is the line of symmetry?

c.

What is the distance from line of symmetry to an $x$ -intercept?

Ready for More?

Find two methods for solving this equation graphically:

$3 x^{2} + 3 x - 5 = - x + 7$

Takeaways

Solving quadratic equations using inverse operations:

Example	Procedure
${(x + 2)}^{2} = 16$	Given

Solving quadratic equations by graphing:

Example	Procedure
${(x + 2)}^{2} = 16$	Given

Solving quadratic equations by factoring:

Example	Procedure
$x^{2} + 4 x - 12 = 0$	Given

Solving quadratic equations by completing the square:

Example	Procedure
$x^{2} + 4 x - 12 = 0$	Given

Vocabulary

zero property of multiplication (also called the zero product property)
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned methods for solving quadratic equations. Some quadratic equations can be solved using inverse operations and taking the square root of both sides of the equations. Some quadratic equations can be solved by factoring and using the zero product property. Some quadratic equations can be solved by completing the square and then using inverse operations. Quadratic equations that have real solutions can also be solved by graphing, and each of these algebraic methods has connections to graphing.

Retrieval

1.

Find the features for the function represented in the graph.

Intervals of Increase:

Intervals of Decrease:

Maximum:

Minimum:

Domain:

Range :

2.

Use the function to find the indicated values.

$f (x) = x^{2} - 4 x - 32$

a.

$f (0) =$

b.

$f (4) =$

c.

$f (x) = 0, x =$

d.

$f (x) = 13, x =$