%5E2-(x%2B5)%5E2%3D0.jpeg "(2x-3)^2-(x+5)^2=0")
Solving the Equation (2x-3)^2 - (x+5)^2 = 0
This equation presents a classic algebraic problem that can be solved using a variety of methods. Let's explore some of the most common approaches:
1. Factoring using the Difference of Squares
The equation is in the form of a² - b² = 0, where:
- a = 2x - 3
- b = x + 5
We can utilize the difference of squares factorization: a² - b² = (a + b)(a - b).
Applying this to our equation:
(2x - 3)² - (x + 5)² = (2x - 3 + x + 5)(2x - 3 - x - 5) = 0
Simplifying further:
(3x + 2)(x - 8) = 0
Therefore, for the product to be zero, either:
- 3x + 2 = 0 => x = -2/3
- x - 8 = 0 => x = 8
The solutions to the equation are x = -2/3 and x = 8.
2. Expanding and Solving the Quadratic Equation
Alternatively, we can expand the equation and solve it as a standard quadratic equation:
(2x - 3)² - (x + 5)² = 0 4x² - 12x + 9 - (x² + 10x + 25) = 0 3x² - 22x - 16 = 0
We can now use the quadratic formula to solve for x:
x = [-b ± √(b² - 4ac)] / 2a
Where:
- a = 3
- b = -22
- c = -16
Substituting the values and solving, we again get the solutions:
- x = -2/3
- x = 8
Conclusion
Both methods demonstrate how to solve the equation (2x-3)² - (x+5)² = 0. The factoring approach offers a more efficient and elegant solution in this case, while expanding and using the quadratic formula is a more general method that can be applied to a wider range of quadratic equations.