Question

Solve each equation. Check for extraneous solutions. √5 x+1 -2√x=1

Answer 1

The solution of the equation √5 x + 1 - 2√x = 1 is $\frac{4}{5}$.

An extraneous solution is one that results from the process of addressing the problem but is not a legitimate solution to the problem, such as the answer to an equation.

Consider the equation,

√5 x + 1 - 2√x = 1

Subtracting 1 from each side of the equation.

√5 x + 1 - 2√x - 1 = 1 - 1

√5 x - 2√x = 0

√5 x = 2√x

Dividing each side of the equation by √x.

( √5 x/√x ) = ( 2√x / √x )

√(5x) = 2

Squaring each side of the equation.

[ √(5x) ]² = (2)²

5x = 4

Dividing by 5 from each side of the equation.

( 5x/5 ) = ( 4/5)

x = 4/5

Now checking the extraneous solution:

$\sqrt{5}x + 1 -2\sqrt{x} =1\\\sqrt{5}(\frac{4}{5} ) + 1 - 2(\sqrt{\frac{4}{5} } ) = 1\\ \frac{4}{\sqrt{5} } + 1 - \frac{4}{\sqrt{5} } = 1\\ 1 = 1$

The solution for the equation is 4/5.

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