Algebra Demystified 2nd Ed

364 alGEbra De mystif ieD

equation ax^2 + bx + c = 0 can be written in the form ()x±=number^2

another number. Taking the square root of each side of this equation and

then solving for x gives us the solutions

x bbac

a

bbac

a

=−+ −−−−

(^224)
2
4
2
,
These solutions are abbreviated as
x bbac
a
=−± −
(^24)
2
This formula is called the quadratic formula. It is very important in algebra and
is worth memorizing. You might wonder why we bother factoring quadratic
expressions to solve quadratic equations when the quadratic formula solves any
quadratic equation. There are two reasons. One, factoring is an important skill
in algebra and calculus, used to solve a variety of problems. Two, the factoring
method is often easier and faster to use than computing the quadratic formula.
We normally use the quadratic formula to solve quadratic equations where the
factoring is difficult. For now, we will learn how to evaluate the quadratic for-
mula and simplify the numbers that we get from it. We then use it to solve
quadratic equations.
Before the formula can be used, the quadratic formula must be in the form
ax^2 + bx + c = 0. Once a, b, and c are identified, applying the quadratic formula
is simply a matter of performing arithmetic and simplifying the solutions. We
begin by identifying a, b, and c.
2 x^2 − x −7 = 0 x = 2, b = −1, c = −7
10 x^2 − 4 = 0 is
equivalent to
l0x^2 + 0x − 4 = 0
a = 10, b = 0, c = −4
3 x^2 + x = 0
is equivalent to
3 x^2 + x + 0 = 0
a = 3, b = 1, c = 0
4 x^2 = 0
is equivalent to
4 x^2 + 0x + 0 = 0
a = 4, b = 0, c = 0
x^2 + 3x = 4
is equivalent to
x^2 + 3x − 4 = 0
a = 1, b = 3, c = −4
−8x^2 = −64
is equivalent to
8 x^2 + 0x − 64 = 0
a = 8, b = 0, c = −64