How to Solve the Exponential Equation (6 – x)^(x² + 7x + 12) = 1

Exponential math problem: How to solve the exponential equation (6 - x) raise to (x² + 7x + 12) equal to 1

Solve the exponential equation (6 – x)^(x² + 7x + 12) = 1

Solution to the exponential equation

Solution of this equation involve three different cases,

Case 1: 6 – x = 1

Case 2: 6 – x = -1 when x² + 7x + 12 = even number

Case 3: x² + 7x + 12 = 0 when 6 – x ≠ 0

Case 1

6 – x = 1

⇒ x = 5

Case 2

6 – x = -1

⇒ x = 7

Now check the answer by plugging the solution in x² + 7x + 12

x² + 7x + 12 = 7² + 7•7 + 12 = 49 + 49 + 12 = 110

⇒ x² + 7x + 12 is an even number

⇒ x = 7 is a solution

Case 3

x² + 7x + 12 is a quadratic equation, so we can solve this equation using factorization

x² + 7x + 12 = 0

⇒ x² + 3x + 4x + 12 = 0

⇒ x(x + 3) + 4(x + 3) = 0

so, (x + 3) (x + 4) = 0

thus, x = -3, -4

Now check the solution by plugging the solution in 6 – x

When x = -3

6 – x = 6 – (-3) = 9

6 – x ≠ 0, so x = -3 is a solution

When x = -4

6 – x = 6 – (-4) = 10

6 – x ≠ 0, so x = -4 is a solution

so we got x = 5, 7, -3 & -4

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