Algebra Demystified 2nd Ed

350 alGEbra De mystif ieD

Solving Quadratic Equations by Factoring

A quadratic equation is one that we can write in the form ax^2 + bx + c = 0

where a, b, and c are numbers and a is not zero (b and/or c might be zero). For

instance, we can write the equation 3x^2 + 7x = 4 in the above form, so it is a

quadratic equation.

374

44

3740

2

2

xx

xx

+=

−−

+−=

We use one of two main approaches to solve quadratic equations. One approach

is based on the fact that if the product of two numbers is zero, then at least one

of the numbers must be zero. In other words, wz = 0 implies w = 0 or z = 0 (or

both w = 0 and z = 0). To use this fact on a quadratic equation we first make sure

that one side of the equation is zero. Once this is done, we factor the nonzero side.

We then set each factor equal to zero and solve one or two linear equations.

EXAMPLES

Solve the equation by factoring.

x^2 + 2x – 3 = 0

One side of the equation is already 0, so we can begin by factoring the

nonzero side. x^2 + 2x – 3 factors as (x + 3)(x – 1), so x^2 + 2x – 3 = 0 becomes

(x + 3)(x – 1) = 0.

We now set each factor equal to zero and solve for x.

xx

x

+  =  = 

+  +

30 10

33 11

−

−−

=== − 31 x

We can check our solutions by substituting them into the original equation,

xx

x

x

2

2

2

230

332339630

11 21

+−=

=− −+−−=−−=

=+

:() ()

: ()



−−=+−= 31230 

EXAMPLES

Solve the equation by factoring.

x

EXAMPLES

Solve the equation by factoring.