Question

How do you solve `6x^2 - 27x +27= 0` by factoring?

Answer 1

The solutions are
`color(blue)(x=3/2`

`color(blue)(x=3`

Explanation:

`6x^2−27x+27=0`

We can Split the Middle Term of this expression to factorise it and thereby find solutions.

In this technique, if we have to factorise an expression like `ax^2 + bx + c`, we need to think of 2 numbers such that:

`N_1*N_2 = a*c = 6*27 = 162`
and
`N_1 +N_2 = b = -27`

After trying out a few numbers we get `N_1 = -18` and `N_2 =-9`
`-18*-9 = 162`, and `(-18)+(-9)=-27`

`6x^2−27x+27=6x^2−18x-9x+27`
`6x(x-3) -9(x-3) =0`

`(6x-9)(x-3) =0` is the factorised form of the expression.

We now equate each of these factors to the R.H.S (`0`)
`6x-9=0, x=9/6,color(blue)(x=3/2`
`x-3=0, color(blue)(x=3`