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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 ax2 + bx + c = 0

Solve the equation

Step Then …
x2 + 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 = x2. 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 = x2 + 1

Notice that this is just the graph y = x2 moved up
by 1.

3. Graph y = x2 - 2

4. Graph y = - x2

5.

6.

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