How do I find the values for a, b, and c, to complete the solution?

Answer:

The values a, b, and c are: a = - 4, b = 3, c = 6

The solutions to the equation are: x = - 4 + 3√6 and x = - 4 - 3√6

Explanation:

1) First step is complete squares:

x² + 8x = 38 ← given

x² + 8x + (8/2)² = 38 + (8/4)² ← add square of the half of 8

x² + 8x + 16 = 38 + 16 ← solve the parenthesis

x² + 8x  + 16 = 54 ← combine like terms

2) Second, solve the equation:

(x + 4)² = 54 ← factor the perfect square trinomial

(x + 4) = +/- √54 ← extract square root of both sides

x = - 4 +/- √54 ← subtract 4 from both sides

x = -4 +/- 3√6 ← simplify the root

x = - 4 + 3√6 and x = - 4 - 3√6 ← separate the two solutions

3) Third, compare with the solutions shown:

x = a + b√c and x = a - b√c

⇒ a = -4, b = 3, c = 6