Solving quadratic equations by FACTORING

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A quadratic equation always equals zero, and always has a squared term. A properly arranged quadratic equation is modeled like this:

ax² + bx +c = 0

'a', 'b' and 'c' are all coefficients (numbers) and they may be positive OR negative. Here is an example:

4x² - 17x -15 = 0

HOW TO SOLVE A QUADRATIC EQUATION

There are many ways to solve a quadratic equation. The three main ways are completing the square, using the Quadratic Formula and factoring. This lesson demonstrates how to use factoring to solve a quadratic equation.

FACTORING

Multiply the 'a' coefficient by the 'c' coefficient. Ignore signs for now.

4 ('a' coefficient) x 15 ('c' coefficient) = 60

Find the factors of 60 ('what' times 'what' equals 60), looking for a pair that equal -17 (the 'b' coefficient) when added together:

Factors of 60: [1, 60] : [2, 30] : [3, 20] : [4, 15] : [5, 12] : [6, 10] 

Out of all of these pairs, only one pair works, and that is only if you have the signs in a specific order:

-20 + 3 = -17

Now you set up two fresh binomials with our 'a' and 'c' terms, which look like this:

(4x² +      ) and (      - 15)

You then take the two numbers that you factored, and put them, combined with an 'x', in the missing places in the binomial model. It doesn't matter which one you put where.

(4x² - 20x) and (3x -15)

Now you factor out the common terms, also known as "Reverse Distribution":

4x(x-5) and 3(x-5)

If the new terms in the binomials match EXACTLY, then that is one of your solution sets. In our example, both sets now contain exactly (x-5). Combine the factors on the outside of the binomials. This is your second solution set:

4x(x-5) and 3(x-5)

(x-5) (4x+3)

Use the zero property to solve for 'x' for both sets:

x-5 = 0

x = 5

4x+3 = 0

4x = -3

x = -3/4

Your solution is x = 5 and -3/4. This means that if you put either number into our original equation, you will make it equal zero. These two points are also where the graphed parabola crosses the x-axis. These are also known as ‘roots’. The ‘solutions’ are the actual ‘x’ values for the coordinates that cross the x-axis: (5, 0) and (-3/4, 0)

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