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The Humongous Book of Algebra Problems by W. Michael Kelley

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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 cant
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 theres
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
doesnt 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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