Question

How do you solve `5x^2 - 7x - 6 = 0` by factoring?

Answer 1

The solutions are

`color(blue)(x=-3/5`
`color(blue)(x=2`

Explanation:

`5x^2−7x−6=0`

We can Split the Middle Term of this expression to factorise it and thereby find the 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 = 5*-6 = -30`
and
`N_1 +N_2 = b = -7`

After trying out a few numbers we get `N_1 = -10` and `N_2 =3`
`-10*3 = -30`, and `3+(-10)= -7`

`5x^2−color(blue)(7x)−6= 5x^2−color(blue)(10x +3x)−6`

`= 5x(x-2) + 3(x-2)`

`color(blue)((5x+3)(x-2)` are the factors of the expression.

Now we equate these factors to zero and find the solutions:

`5x+3 = 0 , color(blue)(x=-3/5`
`x-2 =0, color(blue)(x=2`