Question

How do you factor and solve `3x^2+5x+2=0`?

Answer 1

The solutions for the equation are

`color(green)(x = -2/3`

`color(green)(x = -1`

Explanation:

`3x^2 + 5x + 2 = 0`

We can Split the Middle Term of this expression to factorise it.

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 = 3*2 = 6`

AND

`N_1 +N_2 = b = 5`

After trying out a few numbers we get `N_1 = 3` and `N_2 =2`
`3*2 = 6`, and `3 + 2= 5`

Factorising the expression:

`3x^2 +color(blue)( 5x) + 2 = 3x^2 + color(blue)(3x + 2x) + 2`

`= 3x(x + 1 ) + 2 ( x + 1 )`

`(x +1 )` is a common factor to each of the terms

`=color(blue)((3x + 2 ) ( x + 1 )`

We now equate the factors to zero to obtain the solutions:

• `3x + 2 = 0, color(green)(x = -2/3`

• `x +1 = 0, color(green)(x = -1`