The Factoring method for solving Quadratic Equations
Here are three general types of polynomial equations in one variable ( a 0 ):
Linear: ax b 0
Quadratic: ax2 bx c 0
Cubic: ax3 bx2 cx d 0
The ax2 term is called the quadratic term, the bx term is the linear term, and the c is the
A root, or solution, of a polynomial equation is a value of the variable that satisfies the
For example, the roots of x 2 5x 24 0 are -3 and 8, because when you plug in either
of these values for x, the equation holds.
When a polynomial equation in x is written with 0 as one side, one way to solve the
equation is by factoring the other side into linear factors of the form ax+b.
This method of solving quadratic equations is called the factoring method, and relies on
the zero-product property.
The zero-product property states that ab 0 if an only if a 0 or b 0 .
To use the zero-product property to solve a polynomial equation,
1. write the equation with 0 as one side,
2. factor the other side of the equation, and
3. solve the equation obtained by setting each factor equal to 0.
Solve: x 2 x 30
x 2 x 30 0
( x 5)(x 6) 0
x5 0 x6 0
x 5 x6
A number r is a zero of a function f if f(r) = 0. If one factor occurs twice as a factor of f,
this is called a double root.
For example, x( x 2) 2 has three roots, x, (x-2), and (x-2). Since (x-2) occurs twice, it is
called a double root.
Solving Quadratics by Factoring Practice problems: Score: /5
Solve. Identify all double roots.
( x 1)(x 4) 0
z 2 3 4z
3r 2 4r 1
10t 2 9t 1
(u 3)(u 3) 8u
** ( x 1)3 ( x 1) 2 0
** A graphic artist is designing a poster that consists of a rectangular print with a uniform
border. The print is to be twice as tall as it is wide, and the border is to be 3 in. wide. If
the area of the poster is to be 680 in2 , find the dimensions of the print.