# Exponential Equations Reducible to Quadratics

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

### Question 1

Solve $9^x-10 \times 3^x + 9 = 0$.

\begin{aligned} \displaystyle \require{AMSsymbols} \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}

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