Chapter Fourteen — Quadratic Equations and Inequalities

The Humongous Book of Algebra Problems

296

Solving Quadratics by Factoring

Use techniques from Chapter 12 to solve equations

14.1 Solve the equation: xy = 0.

According to the zero product property, a product can only equal 0 if at least

one of the factors (in this case either x or y) equals 0. Therefore, xy = 0 if and

only if x = 0 or y = 0.

14.2 Solve the equation: x(x – 3) = 0.

The product of x and (x – 3) is equal to 0. According to the zero product

property (explained in Problem 14.1), one (or both) of the factors must equal 0.

x = 0 or x – 3 = 0

Solve x – 3 = 0 by adding 3 to both sides of the equation.

x = 0 or x = 3

The solution to the equation x(x – 3) = 0 is x = 0 or x = 3.

14.3 Solve the equation: (x + 2)(2x – 9) = 0.

According to the zero product property, the product left of the equal sign only

equals 0 if either (x + 2) or (2x – 9) equals 0. Set both factors equal to 0 and

solve the equations.

The solution to the equation (x + 2)(2x – 9) = 0 is x = –2 or .

Note: Problems 14.4–14.5 demonstrate two different ways to solve the equation x

2

= 16.

14.4 Solve the equation using square roots.

Problems 13.32–13.36 demonstrate that squaring both sides of an equation that

contains a square root eliminates the root. Conversely, taking the square root

of both sides of an equation containing a perfect square eliminates the perfect

square.

You can’t

multiply two

numbers and

get zero unless one

(or both) of those

numbers is 0. That’s a

property that is unique

to 0. For example, if

xy = 4, then there’s

no guarantee that

either x = 4 or

y = 4. You could set

x = 2 and y = 2, or

maybe x = 8 and

y = 0.5.

There

are two

possible

solutions to |

the equation.

Use the word

“or” to separate

them because

plugging either

x = 0 OR x = 3 into

the equation produces

a true statement.

Mathematically,

saying “x = 0 AND

x = 3” means x has to

equal both of those

things at the same

time, and that

doesn’t make

sense.

Back in Chapter 13, you only squared

both sides AFTER you isolated the square

root on one side of the equal sign. Similarly,

only square root both sides when the perfect

square is isolated on one side of the equal sign.

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