Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

8.4.2 Solving Equations of Increasing Complexity Using Reasoning

8.4.2 • Solving Equations of Increasing Complexity Using Reasoning

Activity

For each equation, find its solution or solutions. Be prepared to explain your reasoning.

For questions 1 – 3, review the process of solving equations.

1.

$x - 3 = 0$

Compare your answer:

$x = 3$

2.

$x + 11 = 0$

Compare your answer:

$x = - 11$

3.

$2 x + 11 = 0$

Compare your answer:

$x = \frac{- 11}{2}$

For questions 4 – 7, recall the zero product property means that each (or all) of the factors could be equal to zero when their product is zero. Solve each equation by setting each factor equal to zero and solving.

4.

$x (2 x + 11) = 0$

Compare your answer:

$x = 0$ and $x = \frac{- 11}{2}$

5.

$(x - 3) (x + 11) = 0$

Compare your answer:

$x = 3$ and $x = - 11$

6.

$(x - 3) (2 x + 11) = 0$

Compare your answer:

$x = 3$ and $x = \frac{- 11}{2}$

7.

$x (x + 3) (3 x - 4) = 0$

Compare your answer:

$x = 0$ , $x = - 3$ , and $x = \frac{4}{3}$

Are you ready for more?

Extending Your Thinking

1.

Use factors of 48 to find as many solutions as you can to the equation $(x - 3) (x + 5) = 48$ .

Compare your answer:

$x = 7$ and $x = - 9$

2.

Once you think you have all the solutions, explain why these must be the only solutions.

Compare your answer:

The numbers expressed by $x - 3$ and $x + 5$ are 8 units apart. The only factor pairs of 48 that are 8 units apart are 4 and 12, as well as –4 and –12.

Think:

What value of $x$ would make the factors $(x - 3) (x + 5)$ equivalent to $(4) (12)$ ? $x = 7$ .
What value of $x$ would make the factors $(x - 3) (x + 5)$ equivalent to $(- 4) (- 12)$ ? $x = - 9$ .

Check:

If $x = 7$ , then $(x - 3) (x + 5) = 4 \cdot 12$ .
If $x = - 9$ , then $(x - 3) (x + 5) = - 12 \cdot (- 4)$ .

Self Check

Solve the equation.

$(x+4)(8x−16)=0$

$x=2$ and $x=-4$
$x=2$ and $x=-2$
$x=4$ and $x=-2$
$x=4$ and $x=-4$

Additional Resources

Solving Equations of Increasing Complexity Using Reasoning

Let’s look again at the zero product property.

ZERO PRODUCT PROPERTY

If the product of two expressions is 0, then one or both of the expressions equal 0.

If $a b = 0$ , then $a = 0$ or $b = 0$ , or $a = b = 0$ .

We can use the zero product property to help reason through finding the solutions to complex quadratic equations.

We know that if factors are multiplied together and the result is 0, then one of those factors must equal 0.

Example 1

$(x - 7) (2 x + 12) = 0$

This equation has two different factors that, when multiplied, result in a product of 0.

To find possible solutions to this equation, we can set each factor equal to 0 and solve.

$(x - 7) = 0 (2 x + 12) = 0$

The value $x = 7$ makes the first factor equal to 0. The value $x = - 6$ makes the second factor equal to 0.

So, the solutions to the original equation are $x = 7$ and $x = - 6$ .

Example 2

$x (x - 9) (3 x - 8) = 0$

This equation has three different factors, but the reasoning is the same. We can still set each factor equal to 0 and solve.

The first factor is just $x$ . So, $x = 0$ is a solution.

The second factor gives a solution of $x = 9$ when we solve $(x - 9) = 0$ .

The third factor gives a solution of $x = \frac{8}{3}$ when we solve $(3 x - 8) = 0$ .

So, the solutions to $x (x - 9) (3 x - 8) = 0$ are $x = 0$ , $x = \frac{8}{3}$ , and $x = 9$ .

Try it

Solving Equations of Increasing Complexity Using Reasoning

Use the zero product property to reason through finding the solutions to the following quadratic equations.

1. $(x - 6) (4 x + 20) = 0$

Think about which values make each factor equal to 0.

$x = 6$ and $x = - 5$

Use the zero product property to reason through finding the solutions to the following quadratic equations.

2. $(x + 1) (5 x + 14) = 0$

Think about which values make each factor equal to 0.

$x = - 1$ and $x = \frac{- 14}{5}$