Final answer:
To solve the equation 5(x-3)² + 4 = 129, first distribute the 5 to the terms inside the parentheses, then expand the equation by squaring the binomial, and simplify. Then, rearrange the terms in standard form and factor or use the quadratic formula to find the solutions. The solutions to the equation are x = 8 and x = -2.
Step-by-step explanation:
- Simplify the equation by distributing the 5 to the terms inside the parentheses: 5(x-3)² + 4 = 129.
- Expand the equation further by squaring the binomial inside the parentheses: 5(x²-6x+9) + 4 = 129.
- Distribute the 5 to all terms in the parentheses: 5x² - 30x + 45 + 4 = 129.
- Combine like terms: 5x² - 30x + 49 = 129.
- Move the constant term to the other side of the equation by subtracting 49 from both sides: 5x² - 30x = 80.
- Divide the entire equation by the coefficient of x², which is 5: (5x² - 30x)/5 = 80/5.
- Simplify the left side of the equation: x² - 6x = 16.
- Rewrite the equation in standard form by rearranging the terms: x² - 6x - 16 = 0.
- Factor the quadratic equation or use the quadratic formula to find the solutions. In this case, the equation can be factored as (x - 8)(x + 2) = 0.
- Set each factor equal to zero and solve for x: x - 8 = 0 or x + 2 = 0.
- Solve for x in each equation: x = 8 or x = -2.