Math 101 Intermediate Algebra

Solving Quadratic Equations Using Factoring
Chapter 6, Section 5

This section covers using factoring to algebraically solve for x equations having the form

ax2 + bx + c = 0,  a not zero,

which is the standard form of a quadratic equation in one variable.

Zero-Factor Property

For all real-valued factors a and b
if a · b = 0,
then one (or both) of a and b must be 0.

To Solve a Quadratic Equation in One Variable by Factoring

  1. Get 0 on one side of the "=" sign and everything else on the other side.

  2. Simplify by combining like terms (on the side of the equation without the 0).

  3. Factor (the side of the equation without the 0).

  4. Set to 0 each factor containing the variable.
    (This is application of the zero-factor property.)

  5. To determine the solutions, solve each equation obtained in step 4.

  6. Check the solutions in the original equation.

All quadratic equations have 2 solutions.

Sometimes the 2 solutions have the same value.

Not all quadratic equations have real-valued solutions.

Finding the Intercepts of the Graph of a Quadratic Equation

Considering the graph of y = ax2 + bx + c,

  • To find the x-intercepts
    • Set y = 0 and solve for x.
    • That means, solve 0 = ax2 + bx + c.
    • The x-intercepts are (x1, 0) and (x2, 0), if the solutions are called x1 and x2.

  • To find the y-intercept
    • Set x = 0 and solve for y.
    • That means, y = c.
    • The y-intercept is (0, c).