An Exponential Equation

Algebra Level 3

Solve the equation 9 x + 1 + 3 1 − 2 x = 28 9^{\sqrt{x}+1}+3^{1-2\sqrt{x}}=28 .


The answer is 0.25.

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1 solution

Using the Laws of Indices, 9 x + 1 + 3 1 − 2 x = 28 9^{\sqrt{x}+1}+3^{1-2\sqrt{x}}=28 becomes 9\times{9^\sqrt{x}}+3\times{\frac{1}{3^{2\sqrt{x}}}}=28 . Letting q=9^\sqrt{x}\geq{1} and simplifying further gives 9 q + 3 q = 28 9q+\frac{3}{q}=28 . 9 q 2 − 28 q + 3 = 0 ⟹ ( 9 q − 1 ) ( q − 3 ) = 0 ⟹ q = 1 9 9q^2-28q+3=0\implies{(9q-1)(q-3)=0}\implies{q=\frac{1}{9}} (rejected) or q = 3 q=3 Now we know that 9^\sqrt{x}=3\implies{\sqrt{x}=\frac{1}{2}}\implies{x=\frac{1}{4}} . ∴ x = 0.25 \therefore{x=0.25}

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sqrt(x)=-1 i.e. x=1 also satisfies the given equation This is directly from q=1/9 as you've solved above; don't reject it.

vinayak nayak - 5 years, 2 months ago

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