Solving Quadratic Equations

(what we've all been waiting for, right?)

So, you want to solve a quadratic equations.  Well, a quadratic equation in standard form is written:
ax² + bx + c = 0

You may recognize this as a quadratic trinomial set equal to 0, which it is.  One way to solve an equation like this is by factoring.  It may not always work, because the expression equal to 0 is not always factorable.  Let's take one that is:

x² + 5x + 6 = 0

Now, we know how to factor the left side of the equation, so we do so:

(x + 2)(x + 3) = 0

By a property called the Zero Product Rule, if ab = 0, then a = 0, b = 0, or both  = 0.

Using this, we can give a the value of x + 2, and b the value of x + 3.

Therefore, x + 2 = 0 OR x + 3 = 0
x = -2 OR x = -3

Substitute both values back in to make sure they work:
(-2)² + 5(-2) + 6 = 0
4 - 10 + 6 = 0
10 - 10 = 0 YES
(-3)² + 5(-3) + 6 = 0
9 - 15 + 6 = 0
15 - 15 = 0 YES

Let's do another one:
x² - 8x + 15 = 0
(x - 3)(x - 5) = 0
x - 3 = 0 OR x - 5 = 0
x = 3 OR x = 5

We'll do one more, that is a little harder to factor.
3x² + 7x + 2 = 0
(3x² + 6x) + (x + 2) = 0
3x(x + 2) + 1(x + 2) = 0
(3x + 1)(x + 2) = 0
3x  + 1 = 0 OR x + 2 = 0
x = -1/3 OR x = -2

Now we will look at some special factoring patterns:
(a + b)(a - b) = a² - b²
(a + b)² = a² + 2ab + b²

For the next way to solve quadratic equations, we will concentrate on using the second one.  This process is called completing the square.

Take x² + 8x + 12 = 0
What we want is to make the expression on the left into the form that can be factored into (a + b)².  Therefore, we need to change c (the constant).  First we subtract it from both sides.

x² + 8x + 12 = 0
              -12   -12
x² + 8x = -12

Then, with this different form, we add the number needed to complete the square on the left side, to both sides.  We have to think...
In the factoring pattern, x² would have to equal a², so x = a.  8x = 2ab.  8x = 2xb.  Divide each side of that equation by 2x and get 4 = b.  Because we want the constant we add to be b², we add 16 to both sides.  Does this make sense?

x² + 8x + 16 = -12 + 16
(x + 4)² = 4 (We factored the left side according to the pattern, and simplified the right side.)
Now take the square root of both sides, but don't forget that on the right it can be positive OR negative.
x + 4 = ±√4
x + 4 = ±2
x = -2 OR x = -6

Try another: x² + 6x + 11 = 0
x² + 6x = -11
**x = a, 6x = 2xb, b = 3, b² = 9**
x² + 6x + 9 = 11 + 9
(x + 3)² = 20
x + 3 = ±√20
x = ±√20 - 3
What that translates to is:
x = √20 - 3 OR x = -√20 -3

By the way, I'm really sorry that the lines over the radicands don't come out right (I had to underline the line above them).  If you are not exactly sure what is inside and what is outside the radicand, use your common sense and bear with me (please).

Also, if a quadratic equation is ever not in standard form, change it to it by adding or subtracting terms.

Click here to go to the next page of solving quadratic equations (the quadratic formula).
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