Quadratic equations

There are different methods for solving quadratic equations.

1. 1.      Factorisation:

To solve x² + 5x + 6 = 0 by factorisation:

x² + 5x + 6 = 0 factorised is (x+2)(x+3)=0

You know this because x² is x multiplied by itself. The last terms in each of the brackets must add up to 5 and multiply to give 6.

Everything in the bracket is considered as one term, and the brackets are the factors of the equation the came from. So saying (x+2) is a factor of x² + 5x + 6 = 0 is the same as saying 2 is a factor of 6. Everything in a bracket is treated as one term.

So to factorise x² + 7x + 12 = 0, you know x² is x multiplied by x. When factorising a quadratic, one of the signs is usually the same as the first sign in the equation, so it’s (x+ )(x+ )=0 so far. In the first bracket, it’s a plus because the first sign in the equation is a plus and you need to work out what is needed to make the second sign in the equation. In this equation, it’s also a plus, so the second sign in the factorisation should be a plus as well. The answer is (x+3)(x+4)=0.

So to solve this question, you know that (x+3) and (x+4) are two factors of the equation x² + 7x + 12 = 0. The equation has 0 on the right hand side and so you know that (x+3) multiplied by (x+4) should equal 0, you also know that anything multiplied by 0 is 0, so one of the factors must equal 0. If (x+3) is equal to zero that means it must be (-3+3)(x+4).

X is -3  and -4.

Not all quadratics can be factorised, so another method is

1. 2.      Completing the square

Some quadratics can be changed to become a perfect square. E.g. x² – 6x + 2 = 0.

Subtract the constant (which is -2) from both sides:
x² – 6x + 2 = 0
x² – 6x + 2(-2) = 0 (-2)

x² – 6x  = -2

Take the coefficient of x (that’s the -6 in front of x) and halve it: half of -6 is -3.

Square -3, this results in 9, add 9 to both sides of x² – 6x  = -2 and this is:
x² – 6x+ 9 = 7, factorising gives (x-3)²  = 7. Make sure the right hand side is 0, (x-3)² – 7  = 0  <- – -   Completed square form.

1. 3.      Quadratic Formula

The quadratic formula is  . To use the quadratic formula, you only need to substitute the values, after making sure that the quadratic equation is in the general form of ax + bx + c = 0.