This week we learned about quadratic equations.

There are three main ways to solve quadratic equations:

1. Factoring

2. Completing the Square

3. Quadratic Formula

Factoring:

Factoring quadratics is very similar to factoring polynomials, except there are two solutions instead of just one.

Here’s an example:
x^2 + 9x + 20 = 0

(x + 5) ( x + 4) = 0

Then from this point, you must find what each x must equal to make the left side equal zero. So:

x = -5, x = -4

Completing the Square: 

Completing the square is helpful when factoring isn’t possible.

Here’s an example:

x^2 + 10x +8 = 0

Since there is no way to factor this nicely, what you do is take half of the second term (10x), and square it to make a perfect square trinomial:

x^2 + 10x + - + 8 = 0

Note: the numbers will be added in as a zero pair so the equation will still be equivalent.

x^2 + 10x + 25 - 25 + 8 = 0

You then must identify the perfect square trinomial and factor it:

(x + 5)^2 - 25 + 8 = 0

Then find like terms:

(x + 5)^2 -17 = 0

Then isolate x:

(x+5)^2 = 17

 

\sqrt {(x+5)^2} = \sqrt {17}

 

x+5 = +/- \sqrt {17}

 

x = -5 +/- \sqrt {17}

Quadratic Formula:

This is the final method.

It involves using the quadratic formula which is:

x = \frac {-b +/- \sqrt {b^2 - 4ac}}{2a}

When given a quadratic equation like:

x^2 + 6x + 3 = 0

A = the coefficient of the first term

B = the coefficient of the second term

C = the third term

So…

A = 1

B =  6

C = 3

Then just insert the variables into the formula and solve!:

x = \frac {-b +/- \sqrt {b^2 - 4ac}}{2a}

 

x = \frac {-6 +/- \sqrt {6^2 - 4(1)(3)}}{2(1)}

 

x = \frac {-6 +/- \sqrt {36 -12}}{2}

 

x = \frac {-6 +/- \sqrt {24}}{2}

 

x = \frac {-6 + 2 +/- \sqrt {6}}{2}

 

x = -6 +/- \sqrt {6}