Question

Solve for x. Write both solutions, separated by a
comma.
6x² + 5x - 6= 0

Answer 1

`x=(2)/(3),-(3)/(2)`

Explanation:

Given equation:

`6x^2+5x-6=0`

First, factor the left side of the given equation.

To factor a quadratic in the form
`ax^2+bx+c` find two numbers that multiply to
`ac` and sum to
`b`:

`\implies ac=6\cdot-6=-36`

`\implies b=5`

So the two numbers are: 9 and -4

Rewrite
`b` as the sum of these two numbers:

`\implies 6x^2+9x-4x-6=0`

Factorize the first two terms and the last two terms separately:

`\implies 3x(2x+3)-2(2x+3)=0`

Factor out the common term
`(2x+3)`:

`\implies (3x-2)(2x+3)=0`

To solve for x:

`\begin{aligned}\implies (3x-2) & =0 & \implies (2x+3) & = 0\\3x & = 2 & 2x & = -3\\x & = (2)/(3) & x & = -(3)/(2)\end{aligned}`

Therefore:

`x=(2)/(3),-(3)/(2)`