Solve: [tex]\frac{x^2-x-6}{x^2}=\frac{x-6}{2 x}+\frac{2 x+12}{x}[/tex]

After multiplying each side of the equation by the LCD and simplifying, the resulting equation is
[tex]3 x ^2+20 x +12=0[/tex]

What are the solutions to the equation?
A. [tex]x=-6[/tex] and [tex]x=-\frac{2}{3}[/tex]
B. [tex]x=-6[/tex] and [tex]x=\frac{2}{3}[/tex]
C. [tex]x=6[/tex] and [tex]x=\frac{2}{3}[/tex]

Factor the quadratic equation 3 x 2 + 20 x + 12 = 0 .
Rewrite the middle term: 3 x 2 + 2 x + 18 x + 12 = 0 .
Factor by grouping: x ( 3 x + 2 ) + 6 ( 3 x + 2 ) = 0 , which simplifies to ( 3 x + 2 ) ( x + 6 ) = 0 .
Solve for x : x = − 6 and x = − 3 2 ​ . The solutions are x = − 6 and x = − 3 2 ​ ​ .

Explanation

Problem Setup We are given the equation x 2 x 2 − x − 6 ​ = 2 x x − 6 ​ + x 2 x + 12 ​ and told that after multiplying each side of the equation by the LCD and simplifying, the resulting equation is 3 x 2 + 20 x + 12 = 0 . Our goal is to find the solutions to this quadratic equation.

Factoring Preparation To solve the quadratic equation 3 x 2 + 20 x + 12 = 0 , we can use factoring. We look for two numbers that multiply to 3 × 12 = 36 and add up to 20 . These numbers are 2 and 18 .

Rewriting the Middle Term We rewrite the middle term using these numbers: 3 x 2 + 2 x + 18 x + 12 = 0 .

Factoring by Grouping Now, we factor by grouping: x ( 3 x + 2 ) + 6 ( 3 x + 2 ) = 0 .

Factoring out the Common Term We factor out the common term ( 3 x + 2 ) : ( 3 x + 2 ) ( x + 6 ) = 0 .

Setting Factors to Zero We set each factor to zero and solve for x : 3 x + 2 = 0 or x + 6 = 0 .

Solving for x Solving for x , we get 3 x = − 2 , so x = − 3 2 ​ , or x = − 6 . Thus, the solutions are x = − 6 and x = − 3 2 ​ .

Final Answer The solutions to the equation 3 x 2 + 20 x + 12 = 0 are x = − 6 and x = − 3 2 ​ .


Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the growth or decay of populations. For example, if you want to build a rectangular garden with an area of 100 square meters and the length must be 5 meters longer than the width, you can use a quadratic equation to find the dimensions of the garden. Understanding how to solve quadratic equations is essential for solving these types of practical problems.

Answered by GinnyAnswer | 2025-07-03

The solutions to the equation 3 x 2 + 20 x + 12 = 0 are x = − 6 and x = − 3 2 ​ . The correct answer choice is A: x = − 6 and x = − 3 2 ​ .
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Answered by Anonymous | 2025-07-04