Hi, I'll try to explain the basics of quadratic equations, then move on to the formula.


A quadratic equation is an equation in the form

aX^2 + bX + c = 0


these can also be expressed in the form

a(x - p)(x-q) = 0 (as long as a is not zero)

where p and q are the solutions to the equation.

The reason p and q are the solutions is that in the second form we set x = p we get.

a(p - p)(p - q) = 0

as p - p = 0 and anything multiplied by 0 is 0 a(p - p)(p-q) = 0. So p is a solution.

if we say x = q we get

a(q - p)(q - q) = 0

which is true because q - q is 0.

So to solve a quatratic equation all we have to do is find p and q where

a(x - p)(x - q) = aX^2 + bX + c


and p and q are our solutions.


lets look at a simple example where a = 1

x^2 + bx + c = 0 = (x - p)(x - q)

if we multiply out the brackets on the right hand side it'll look a bit like the left hand side.


(x - p)(x - q)

= x(x - q) - p(x - q)

= (x^2 - qx) - (px - pq)

= X^2 - (p + q)x + pq

if we compare this to the original formula

x^2 + bx + c


we see that we can find p and q by equating

-b = p + q
c = pq


so we have to find 2 numbers that add together to make - b and multiply together to make c

so if we try your example

x^2 + 5x + 6 = 0

we just have to find 2 numbers that add together to make - 5 and multiply together to make 6


what about -2 and -3

-2 + -3 = -5
-2 * -3 = 6

so -2 and -3 are the solutions if we put them into the equation


(-2)^2 + 5(-2) + 6
= 4 - 10 + 6
= 0

(-3)^2 + 5(-3) + 6
= 9 - 15 + 6
= 0


and those are the answers.


what about if a isn't 1.

well we have

ax^2 + bx + c = 0

so let's divide both sides by a

(a/a)x^2 + (b/a)x + c/a = 0/a

a/a = 1 and 0/a = 0

so this is just like the simple type of equation.

Try this one.

3x^2 + 18x - 21 = 0


first divide both sides by 3

(3/3)x^2 +(18/3)x - (21/3) = 0/3

so

x^2 + 6x - 7 = 0

so we need 2 numbers that add to make -6 and multiply to make -7

what about 1 and -7

1 + -7 = -6
1 * -7 = -7

so 1 and -7 are the 2 solutions.

check those in the original equation


3*(1^2) + 18*1 - 21 = 0

so 1 is a solution



3*(-7^2) + 18*(-7) - 21

= 3*49 - 18*7 - 21

= 147 - 126 - 21 = 0

so - 7 is a solution.


I've just noticed the sun's coming up so I'd better go to bed, don't worry about the formula for now, Guv or I will explain it later, go through that lot and see if it makes sense, using the formula is easier than factoring, because if you're factoring you still have to guess at the 2 numbers, but you should understand the factoring method, It's more important that you understand how the equations work and why there are 2 solutions. The formula's just something you have to learn.