Final answer:
The solutions to the quadratic equation 2(x + 1)² - 5 = 13 are x = 2 and x = -4.
Step-by-step explanation:
To solve the quadratic equation 2(x + 1)² - 5 = 13, we can follow these steps:
1) Expand the squared term (x + 1)²:
2(x² + 2x + 1) - 5 = 13
2) Distribute 2 to each term inside the parentheses:
2x² + 4x + 2 - 5 = 13
3) Simplify the equation by combining like terms:
2x² + 4x - 3 = 13
4) Move the constant term to the other side of the equation:
2x² + 4x - 3 - 13 = 0
5) Combine like terms:
2x² + 4x - 16 = 0
Now we have a quadratic equation in standard form: ax² + bx + c = 0, where a = 2, b = 4, and c = -16.
To solve the quadratic equation, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Substituting the values of a, b, and c into the formula:
x = (-4 ± √(4² - 4 * 2 * -16)) / (2 * 2)
Simplifying the equation:
x = (-4 ± √(16 + 128)) / 4
x = (-4 ± √144) / 4
x = (-4 ± 12) / 4
Now we have two possible solutions:
a) x = (-4 + 12) / 4 = 8 / 4 = 2
b) x = (-4 - 12) / 4 = -16 / 4 = -4
Therefore, the solutions to the quadratic equation 2(x + 1)² - 5 = 13 are x = 2 and x = -4.