# Exponential Equations Reducible to Quadratics

Exponential Equations Reducible to Quadratic for Math Help is based on various index rules, such as;
$$a^{x+y} = a^x \times a^y \\ (a^x)^y = a^{xy}$$

## Question 1

Solve $9^x – 10 \times 3^x + 9 = 0$.

\begin{aligned} \displaystyle \require{color} (3^x)^2 – 10 \times 3^x + 9 &= 0 &\color{red} 9^x = (3^2)^x = (3^x)^2 \\ (3^x -1)(3^x – 9) &= 0 \\ 3^x = 1 &\text{ or } 3^x = 9 \\ 3^x = 3^0 &\text{ or } 3^x = 3^2 \\ \therefore x = 0 &\text{ or } x = 2 \\ \end{aligned}

## Question 2

Solve $4^x – 2^{x+2} -32 = 0$.

\begin{aligned} \displaystyle \require{color} (2^2)^x – 2^2 \times 2^x – 32 &= 0 \\ (2^x)^2 – 4 \times 2^x – 32 &= 0 \\ (2^x – 8)(2^x + 4) &= 0 \\ 2^x &= 8 &\color{red} 2^x \ne -4 \\ 2^x &= 2^3 \\ \therefore x &= 3 \\ \end{aligned}

## Question 3

Solve $\displaystyle 3^{x^2-3x} = 81$.

\begin{aligned} \displaystyle 3^{x^2-3x} &= 3^4 \\ x^2 – 3x &= 4 \\ x^2 – 3x – 4 &= 0 \\ (x-4)(x+1) &= 0 \\ \therefore x = 4 &\text{ or } x = -1 \\ \end{aligned} \\