In order to solve some quadratic polynomials, factoring will not work.
There is no easy way to factor 3x2 + 3x – 4 = 0. When
this is the case, we turn to the quadratic formula:
Before learning how to use the formula, there are a couple of things
you need to notice about it. First, notice that the b at the beginning
of the formula is negative. This means we are taking the opposite of b.
If b in the equation is already negative, the negative in the formula
still applies.
Second, notice that the formula contains a 
symbol. This gives you the two answers you need because the problem is
a quadratic function.
Let’s take a look at a simple example. This is a problem you could
solve by factoring, but let’s see how it looks in the quadratic
formula.
x2 + 17x + 70 = 0
First, notice that the equation is in ax2 + bx + c = 0. You must
have the equation in this form. Start by identifying the terms to
put into the quadratic formula:
a = 1
b = 17
c = 70
Now, plug these values into the formula.
Simplify the radical.
Reduce and solve.
-10, -7 — these are your answers. Notice that this equation
factors into
(x + 10) (x + 7) = 0.
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Example 2
Not all problems come out so nicely. Here is another example:
3x2 + 12x + 3 = 0
You could try to factor this problem, but you would have trouble.
Turn to the quadratic formula.
a = 3
b = 12
c = 3
Solve what is under the radical first.
Simplify the radical.
Reduce:
This is the answer.
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Example 3
What happens when there are negatives in the equation?
7x2– 5x – 10 = 0
Use the quadratic formula.
a = 7
b = -5
c = -10
Notice that there are two negatives with the five. This is important,
because there is a negative in the formula, and the five in the
equation is also negative.
Simplify the radical.
This is as far as the radical will simplify. However, you cannot
have two negatives with the five. The two negatives use the rules
of multiplication and make a positive.
This is as simple as the equation gets.
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