Solve the equation 2^x - 14*2^-x=-5?show the steps

Wednesday, July 3, 2013

First, we'll re-write the equation, using the negative
power property of exponential functions.

$\displaystyle{{2}}^{{x}}-$
14/(2^x)=-5

We'll multiply by $\displaystyle{{2}}^{{x}}$ both
sides:

$\displaystyle{{2}}^{{2}}{x}-{14}=$
-5*2^x

We'll move all terms to one
side:

$\displaystyle{{2}}^{{2}}{x}+{5}\cdot{{2}}^{{x}}-{14}=$
0

We'll treat the equation above as a quadratic equation
and we'll solve it using quadratic formula, $\displaystyle{{2}}^{{x}}$ being the
unknown:

$\displaystyle{{2}}^{{x}}={\left(-{5}+\sqrt{{{25}+}}\right.}$
56))/2

$\displaystyle{{2}}^{{x}}=$
(-5+sqrt81)/2

$\displaystyle{{2}}^{{x}}=$
(-5+9)/2

$\displaystyle{{2}}^{{x}}={2}$ => x =
1

$\displaystyle{{2}}^{{x}}=\frac{{-{5}-{9}}}{{2}}$

$\displaystyle{{2}}^{{x}}$ = -7
impossible since 2^x > 0 for any real value of
x.

Therefore, the equation will have only one
solution: x = 1.

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