Question

Solve the following exponential equation: 9ˣ² - 3ˣ⁺¹ = 0

Answer 1

$9^{x^2}} - 3^{x + 1} =0$

• Set up

$3^{2 x^2} - 3^{x + 1} = 0$

• Simplify 9 using the fact that $9 = 3^2$

$3^{2 x^2} = 3^{x + 1}$

• Add $3^{x + 1}$ to both sides of the equation

$2 x^2 \log 3 = (x + 1) \log 3$

• Take the $\log$ of both sides

$2 x^2 = x + 1$

• Divide both sides by $\log 3$

$2 x^2 - x - 1 = 0$

• Subtract $x + 1$ from both sides of the equations to create a quadratic equation equaling 0

$(2x + 1)(x - 1) = 0$

• Factor

$2x + 1 = 0 \,\,\textrm{and} \,\, x - 1 = 0$

• Apply the Zero Product Property

$x = - \dfrac{1}{2}, 1$

Our solutions are x = -1/2 and x = 1.