Question

algebra: concepts and connections - plc
solving exponential equations by rewriting the base
solve: ( 12^{x^2 + 5x - 4} = 12^{2x + 6} )
options:
no solution
( x = 2 )
( x = 2, x = -5 )
( x = -5 )

Explanation:

Step1: Set exponents equal

Since the bases are equal ($12
eq 0,1,-1$), we equate the exponents:
$x^2 + 5x - 4 = 2x + 6$

Step2: Rearrange to quadratic form

Subtract $2x+6$ from both sides to standardize:
$x^2 + 3x - 10 = 0$

Step3: Factor the quadratic

Find two factors of $-10$ that sum to $3$:
$(x+5)(x-2) = 0$

Step4: Solve for x

Set each factor equal to zero:
$x+5=0 \implies x=-5$; $x-2=0 \implies x=2$

Answer:

$x=2, x=-5$