(x-2)^2/12-(x+1)^2/21=(x-4)(x-6)/28

3 min read Jun 17, 2024

Solving the Equation: (x-2)^2/12 - (x+1)^2/21 = (x-4)(x-6)/28

This article will guide you through the process of solving the given equation. We will utilize algebraic manipulation to simplify the equation and ultimately find the solutions for x.

1. Finding a Common Denominator

The first step is to find a common denominator for all the fractions in the equation. The least common multiple of 12, 21, and 28 is 84. We can rewrite each fraction with a denominator of 84:

(x-2)^2/12 = 7(x-2)^2/84
(x+1)^2/21 = 4(x+1)^2/84
(x-4)(x-6)/28 = 3(x-4)(x-6)/84

2. Simplifying the Equation

Now, we can rewrite the equation with the common denominator:

7(x-2)^2/84 - 4(x+1)^2/84 = 3(x-4)(x-6)/84

Since all terms have the same denominator, we can eliminate it:

7(x-2)^2 - 4(x+1)^2 = 3(x-4)(x-6)

3. Expanding and Combining Terms

Next, we expand the squares and multiply out the right side of the equation:

7(x^2 - 4x + 4) - 4(x^2 + 2x + 1) = 3(x^2 - 10x + 24)

This simplifies to:

7x^2 - 28x + 28 - 4x^2 - 8x - 4 = 3x^2 - 30x + 72

4. Solving the Quadratic Equation

Combining like terms, we get:

0 = -6x + 48

Solving for x, we have:

x = 8

Conclusion

Therefore, the solution to the equation (x-2)^2/12 - (x+1)^2/21 = (x-4)(x-6)/28 is x = 8.