Question

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

Answer 1

The solutions are
`color(blue)(x=-3`

`color(blue)(x=9`

Explanation:

`x^2−6x−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 =1*-27 = -27`
and
`N_1 +N_2 = b = -6`

After trying out a few numbers we get `N_1 = -9` and `N_2 =3`
`-9*3 = -27`, and `3+(-9)= -6`

`x^2−6x−27=x^2−9x+3x−27`

`x^2−9x+3x−27=0`

`x(x−9)+3(x−9)=0`
`(x+3) (x−9)=0` is the factorised form of the expression.

Now we equate these factors to the R.H.S (`0`)
`x+3=0, color(blue)(x=-3`

`x-9=0, color(blue)(x=9`

Answer 2

Solve y = x^2 - 6x - 27 = 0

Ans: -3 and 9

Explanation:

x^2 - 6x - 27 = 0
Find 2 numbers knowing sum (6) and product (-27). Roots have opposite signs.
Factor pairs of -27 --> (-1, 27)(-3, 9). This last sum is 6 = -b. Therefor, the 2 real roots are: -3 and 9.