(x-7)(x+1)-(x-3)^2=(3x-5)(3x+5)-(3x+1)^2+(x-2)^2-x^2

2 min read Jun 17, 2024

Solving the Equation: (x-7)(x+1)-(x-3)^2=(3x-5)(3x+5)-(3x+1)^2+(x-2)^2-x^2

This article will guide you through the steps to solve the equation (x-7)(x+1)-(x-3)^2=(3x-5)(3x+5)-(3x+1)^2+(x-2)^2-x^2. We will utilize algebraic manipulation to simplify the equation and find the solutions for x.

Step 1: Expanding the Products

We begin by expanding each of the products in the equation.

(x-7)(x+1) = x² - 6x - 7
(x-3)² = x² - 6x + 9
(3x-5)(3x+5) = 9x² - 25
(3x+1)² = 9x² + 6x + 1
(x-2)² = x² - 4x + 4

Substituting these expanded forms back into the original equation, we get:

(x² - 6x - 7) - (x² - 6x + 9) = (9x² - 25) - (9x² + 6x + 1) + (x² - 4x + 4) - x²

Step 2: Simplifying the Equation

Now, we simplify the equation by combining like terms:

x² - 6x - 7 - x² + 6x - 9 = 9x² - 25 - 9x² - 6x - 1 + x² - 4x + 4 - x²
-16 = -10x - 22

Step 3: Isolating x

To isolate x, we add 22 to both sides of the equation:

6 = -10x

Step 4: Solving for x

Finally, we divide both sides by -10 to solve for x:

x = -6/10
x = -3/5

Conclusion

Therefore, the solution to the equation (x-7)(x+1)-(x-3)^2=(3x-5)(3x+5)-(3x+1)^2+(x-2)^2-x² is x = -3/5.