Answer
$x=\left\{ -\dfrac{5}{6},2 \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
First, factor the given equation, $
-6x^2+7x=-10
.$ Then equate each factor to zero and solve for the values of the variable.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the given equation is equivalent to \begin{array}{l}\require{cancel} -6x^2+7x+10=0
\\\\
-1(-6x^2+7x+10)=(0)(-1)
\\\\
6x^2-7x-10=0
.\end{array}
The two numbers whose product is $ac=
6(-10)=-60
$ and whose sum is $b=
-7
$ are $\{
-12,5
\}$. Using these two numbers to decompose the middle term of the equation above results to \begin{array}{l}\require{cancel}
6x^2-12x+5x-10=0
\\\\
(6x^2-12x)+(5x-10)=0
\\\\
6x(x-2)+5(x-2)=0
\\\\
(x-2)(6x+5)=0
.\end{array}
Equating each factor to zero (or the Zero-Factor Property), the solutions to the given equation are \begin{array}{l}\require{cancel}
x-2=0
\\\\
x=2
,\\\\\text{OR}\\\\
6x+5=0
\\\\
6x=-5
\\\\
x=-\dfrac{5}{6}
.\end{array}
Hence, the solutions are $
x=\left\{ -\dfrac{5}{6},2 \right\}
.$
