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An exponential quadratic equation is an equation where the variable we are solving appears in the exponent so that we solve a quadratic equation.

👉This post extends our series on indices 👈

If an exponential equation has the same base on each side of the equation, then the exponent is equal. 

Using this fact, we can solve the following

Example 1
Solve for $x$ in the equation

The base($7$) is the same on both sides, hence the exponents are equal. 

Re-arranging the equation

$x=-2$ or $x=4$

Example 2
Sole for $x$ in the equation

Simplifying  the indicial equation
Let $2^x=t$

By factorization
$t=8$ or $-4$

$t$ can't be negative, hence
$t=8$, but $t≠-4$

Inserting the value of $t$ in $2^x=t$

The base is the same, therefore,

Example 3
Find the value of x in this equation


will translate to

Let $5^x=t$

Taking the L.C.M

Cross multiply

Solving quadratically
$t=1$ or $125$

Substituting the value of t in $5^x=t$
When t=1

When t=125

Therefore $x=3$ and $0$

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Example 4
Solve for x if $2^{2x+1}+2^x-1=0$


Let $2^x=t$

By factorization
$2t-1=0$ or $t+1=0$
$2t=1$ or $t=-1$
$\frac{2t}{2}=\frac{1}{2}$ or $t=-1$
$t=\frac{1}{2}$ or $t=-1$

$t$ cannot be negative, therefore,
$t=\frac{1}{2}$ but $t≠-1$

Inserting the value of t in $2^x=t$

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