2-81%3D0.jpeg "(x-4)2-81=0")
Solving the Quadratic Equation (x-4)² - 81 = 0
This article will guide you through the steps of solving the quadratic equation (x-4)² - 81 = 0.
Understanding the Equation
The equation (x-4)² - 81 = 0 is a quadratic equation in the form of ax² + bx + c = 0. We can identify the coefficients:
- a = 1 (implied by the squared term)
- b = -8 (derived from expanding the square)
- c = -77 (from the constant term -81 + 16)
Solving by Factoring
One way to solve this equation is by factoring. Here's how:
- Recognize the difference of squares pattern: The equation resembles the difference of squares pattern: a² - b² = (a + b)(a - b).
- Apply the pattern: In our case, a = (x-4) and b = 9.
- Factor the equation: (x-4)² - 81 = [(x-4) + 9][(x-4) - 9] = 0
- Solve for x:
- (x-4) + 9 = 0 => x = -5
- (x-4) - 9 = 0 => x = 13
Therefore, the solutions to the equation (x-4)² - 81 = 0 are x = -5 and x = 13.
Solving by the Quadratic Formula
Another way to solve this equation is by using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
- Substitute the values: x = (8 ± √((-8)² - 4 * 1 * -77)) / (2 * 1)
- Simplify: x = (8 ± √(64 + 308)) / 2 = (8 ± √372) / 2
- Calculate the solutions:
- x = (8 + √372) / 2 ≈ 13
- x = (8 - √372) / 2 ≈ -5
This method also gives us the solutions x = -5 and x = 13.
Conclusion
We have successfully solved the quadratic equation (x-4)² - 81 = 0 using two different methods: factoring and the quadratic formula. Both methods lead to the same solutions, demonstrating the versatility of tools available for solving quadratic equations.