Question

solve: $12^{x^2 + 5x - 4} = 12^{2x + 6}$
$\bigcirc$ $x = 2$
$\bigcirc$ $x = -5$
$\bigcirc$ $x = 2, x = -5$
$\bigcirc$ no solution

Explanation:

Step1: Set exponents equal (since bases are equal and base > 0, base ≠ 1)

Since \( 12^{x^2 + 5x - 4}=12^{2x + 6} \), we can set the exponents equal: \( x^2 + 5x - 4 = 2x + 6 \)

Step2: Rearrange into quadratic equation

Subtract \( 2x + 6 \) from both sides: \( x^2 + 5x - 4 - 2x - 6 = 0 \)
Simplify: \( x^2 + 3x - 10 = 0 \)

Step3: Factor the quadratic

Factor \( x^2 + 3x - 10 \): We need two numbers that multiply to -10 and add to 3. The numbers are 5 and -2.
So, \( (x + 5)(x - 2)=0 \)

Step4: Solve for x

Set each factor equal to zero:
\( x + 5 = 0 \) gives \( x = -5 \)
\( x - 2 = 0 \) gives \( x = 2 \)

Answer:

x = 2, x = -5 (Option: x = 2, x = -5)