Equivalent equations are equations with equal values
The solution is: [tex]x = -\frac 23[/tex] or [tex]x = -6[/tex]
The equation is given as:
[tex]\frac{x^2-x-6}{x^2 }= \frac{x-6}{2x} + \frac{2x+12}{x}[/tex]
Take LCM
[tex]\frac{x^2-x-6}{x^2 }= \frac{x-6 + 2(2x + 12)}{2x}[/tex]
[tex]\frac{x^2-x-6}{x^2 }= \frac{x-6 + 4x + 24}{2x}[/tex]
Multiply both sides by x
[tex]\frac{x^2-x-6}{x }= \frac{x-6 + 4x + 24}{2}[/tex]
[tex]\frac{x^2-x-6}{x }= \frac{5x+18}{2}[/tex]
Cross multiply
[tex]2x^2 - 2x - 12 = 5x^2 + 18x[/tex]
Collect like terms
[tex]5x^2 -2x^2 + 18x + 2x + 12 = 0[/tex]
[tex]3x^2 + 20x + 12 = 0[/tex]
Expand
[tex]3x^2 + 18x + 2x + 12 = 0[/tex]
Factorize
[tex]3x(x + 6) + 2(x + 6) = 0[/tex]
Factor out x + 6
[tex](3x + 2)(x + 6) = 0[/tex]
Split
[tex](3x + 2)= 0[/tex] or [tex](x + 6) = 0[/tex]
Solve for x
[tex]x = -\frac 23[/tex] or [tex]x = -6[/tex]
Read more about equivalent equations at:
https://brainly.com/question/19521931
