Question

Solve the following equations:

a) x² - 16x + 64 = 0
b) 2x² - 16x + 32 = 0
c) x² + 8x + 7 = 0
d) x² + 6x - 16 = 0
e) 4 - x² = 2x + 1
f) x² + 2x + 2 = 10
g) 2x(x + 4) = (x + 3)² - 6

Option 1: (-8, -8), Option 2: (4, 4), Option 3: (-7, -1), Option 4: (-8, 2)

Answer 1

To solve the given equations, we can use various methods including factoring and the quadratic formula. Each equation is solved step-by-step, providing the solutions for each equation.

Step-by-step explanation:

To solve each of the given equations, we can apply various methods depending on the form of the equation. Let's go through each equation step-by-step:

1. a) x² - 16x + 64 = 0: This equation can be factored as (x - 8)² = 0, giving us the solution x = 8.
2. b) 2x² - 16x + 32 = 0: Dividing the equation by 2, we get x² - 8x + 16 = 0. Factoring this equation gives us (x - 4)² = 0, giving x = 4 as the solution.
3. c) x² + 8x + 7 = 0: This equation cannot be factored easily, so we can use the quadratic formula. Applying the formula, we get x = (-8 ± √(8² - 4(1)(7))) / (2(1)). Evaluating this expression gives us x ≈ -0.88 and x ≈ -7.12 as the solutions.
4. d) x² + 6x - 16 = 0: Factoring this equation gives us (x + 8)(x - 2) = 0. This means x = -8 or x = 2 are the solutions.
5. e) 4 - x² = 2x + 1: Rearranging this equation gives us x² + 2x - 3 = 0. Factoring this equation gives us (x - 1)(x + 3) = 0. The solutions are x = 1 and x = -3.
6. f) x² + 2x + 2 = 10: Subtracting 10 from both sides gives us x² + 2x - 8 = 0. Factoring this equation gives (x - 2)(x + 4) = 0. So, the solutions are x = 2 and x = -4.
7. g) 2x(x + 4) = (x + 3)² - 6: Expanding the right side gives us x² + 6x + 9 - 6. Simplifying further, we get x² + 6x + 3 = 2x² + 8x. Rearranging this equation gives us 2x² + 2x - 3 = 0. By applying the quadratic formula, we can find the solutions as x ≈ -3.3 and x ≈ 0.45.