Find the values of x for the equation: x^2 - 8x = 105

When presented with an equation that involves x^2, it is likely to be a quadratic equation. This leads us to rearrange to the equation above into the form of ax^2 + b + c = 0. Therefore the equation can be rearranged into x^2 - 8x - 105 = 0.

There are two main methods to solve this quadratic equation - factorising and quadratic formula.

Factorising involves finding two numbers that can multiply together to give -105 and add to form -8. As both numbers are negative we can deduce that one of those numbers are negative and the other is positive.

First start by stating the factors of 105: 1, 3, 5, 7 & 15. Using these numbers, find a pair that would give you a difference of 8. This is 7 and 15. There are two equations that we could form from this:

- (x+7)(x-15)=0

- (x-7)(x+15)=0

Only the first quadratic equation gives rise to -8x and therefore is the correct equation. The values of x is then -7 & 15 as you take the number within the bracket and inverse the sign.

The alternative method is using the quadratic formula: x = (-b +/- SQRT(b^2 - 4ac))/2a. The values for a = 1, b = -8 & c = -105. By substituting the values into the equation we get the answers -7 & 15.