What is the solution set for the equation x² – 8x – 12 = 0?

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To solve the quadratic equation x² – 8x – 12 = 0, we can use the quadratic formula, which is:

  • x = (-b ± √(b² – 4ac)) / 2a

In this formula, a, b, and c are the coefficients from the equation ax² + bx + c = 0. For our equation:

  • a = 1
  • b = -8
  • c = -12

Now, we will plug these values into the quadratic formula:

  1. Calculate the discriminant: b² – 4ac
    • (-8)² – 4(1)(-12) = 64 + 48 = 112
  2. Since the discriminant is positive (112), we will have two real and distinct solutions.
  3. Now plug the values into the quadratic formula:
    • x = (8 ± √112) / 2
  4. We can simplify √112:
    • √112 = √(16 × 7) = 4√7
  5. Now, substituting this back into the formula, we get:
    • x = (8 ± 4√7) / 2
  6. Dividing each term by 2 gives us:
    • x = 4 ± 2√7

Thus, the solution set for the equation x² – 8x – 12 = 0 is:

  • x = 4 + 2√7
  • x = 4 – 2√7

In conclusion, the solution set can be expressed as:{4 + 2√7, 4 – 2√7}.

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