How do I solve the equation x^2 = 3x - 1 ?

First, we need to rearrange the equation into a quadratic. A quadratic equation will always have a squared element. 

To this, we move everything over to the left hand side of the equation and set it equal to zero. (Remember: Change the side, change the sign)
So..
x^2 = 2x - 1

becomes..

x^2 - 2x = - 1
and then..

x^2 - 2x + 1 = 0

One way to solve this equation is by using factorisation.

the equation takes the form.. ax² + bx + c = 0 

we need to find two numbers that add together to make be and multiply together to make a x c.

So, numbers that add to -2 and multiply to 1 x 1 = 1.

These numbers are -1 and -1.

So we can factorise the equation.

Step 1..

x² - x - x - 1 = 0

Step 2..

x(x-1) - 1(x-1) = 0

Step 3..

(x-1)(x-1) = 0

So to find x, we simply divide both sides of the equation by (x-1).

Leaving x - 1 = 0, so x = 1.