What are the solutions to the equation x^2 + 2x - 8 = 0?
Correct Answer: x = -4 and x = 2
- Step 1: Start with the equation x^2 + 2x - 8 = 0.
- Step 2: Look for two numbers that multiply to -8 (the last number) and add to 2 (the middle number).
- Step 3: The numbers that work are 4 and -2 because 4 * -2 = -8 and 4 + (-2) = 2.
- Step 4: Rewrite the equation using these numbers: x^2 + 4x - 2x - 8 = 0.
- Step 5: Group the terms: (x^2 + 4x) + (-2x - 8) = 0.
- Step 6: Factor each group: x(x + 4) - 2(x + 4) = 0.
- Step 7: Factor out the common term (x + 4): (x + 4)(x - 2) = 0.
- Step 8: Set each factor equal to zero: x + 4 = 0 or x - 2 = 0.
- Step 9: Solve for x: From x + 4 = 0, we get x = -4. From x - 2 = 0, we get x = 2.
- Step 10: The solutions to the equation are x = -4 and x = 2.
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