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:

⚡ Using what you learned: Solving Exponential Equations · factoring and solving quadratic equations

Step 1: Equate the exponents

Since the bases on both sides are the same, set their exponents equal to each other:

$$ x^2 + 5x - 4 = 2x + 6 $$

Step 2: Write in standard quadratic form

Subtract \( 2x \) and \( 6 \) from both sides to set the equation to zero:

$$ x^2 + 3x - 10 = 0 $$

Step 3: Factor and solve

Factor the quadratic equation:

$$ (x + 5)(x - 2) = 0 $$

Set each factor to zero:

$$ x + 5 = 0 \implies x = -5 $$

$$ x - 2 = 0 \implies x = 2 $$

Answer:

\( x = 2, x = -5 \)