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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 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 “zerofactor 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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