# Solving Quadratic Equations by F

**Solving Quadratic Equations by Factoring**

A quadratic equation (also called a “second- degree
polynomial ”) is an equation that can be

written as ax^{2} + bx + c = 0

**Solve the equation**

Step … |
Then … |

x^{2} + 3x + 2 = 0 |
Factor the equation |

(x + 2)(x + 1) = 0 | Set each factor equal to 0 to get two equations . |

x + 2 = 0 and x + 1 = 0. | Solve each equation to get two solutions. |

x = -2 and x = -1 | These are the two solutions to the equation . |

Then check the solutions. First we’ll do x = -2.

(-2)^{2} + 3(-2) + 2 = 4 - 6 + 2 = 0. Good. Check the other solution, x
= -1

(-1)^{2} + 3(-1) + 2 = 1 - 3 + 2 = 0. Good. Both solutions check

Comments:

1. It is fine to write: x = -1, -2 or x = -2, -1. It is not OK to write x = (-1,
-2). Those brackets

denote the coordinates of a single point. What we have here are two values for x
that solve the

equation. If you must be fancy you can use set notation x = {-1, -2}

2. Quadratic equations usually have two solutions. (See below for exceptions).

3. The procedure here is quite different from solving equations that don’t have
a squared term in

them. To solve the equation 2x + 5 = 9, the procedure is get x alone on one side
of the equation.

This does not work to solve a quadratic equation.

**Why does it work? **The basic idea is the “zero-factor property,” which
says, “If two numbers

multiply to zero , then one of them must be zero.” That is, if a·b = 0, then
either a = 0 or b = 0.

You can’t multiply two non- zero numbers and get zero.

The point of factoring the quadratic equation is to produce something that looks
like a ·b = 0 so

that we can solve the simple a = 0 and b = 0 equations.

Another example.

The solutions are x = ½, -2

Check:

SPECIAL CASES

1. A perfect square equation has only one solution.

2. The difference of squares .

**PROBLEMS**

** GRAPHING QUADRATIC EQUATIONS
**

1. Graph y = x

^{2}. Let’s pick values for x and find y for each x.

The figure shown is what results from

plotting the points . The curve is called a

parabola .

2. Graph y = x^{2} + 1

Notice that this is just the graph y = x^{2} moved up

by 1.

3. Graph y = x^{2} - 2

4. Graph y = - x^{2}

5.

6.

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