$\frac{5 \sqrt{2 x+5}-1}{\sqrt{2 x+5}+2}$

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Algebra 1

Multiply radical expressions

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\frac{5 \sqrt{2 x+5}-1}{\sqrt{2 x+5}+2}

Identify Expression: Identify the expression to be simplified. $\newline$ We have the expression $(\frac{5\sqrt{2x+5}-1}{\sqrt{2x+5}+2})$ . Our goal is to simplify this expression.
Rationalize Denominator: Rationalize the denominator. $\newline$ To simplify the expression, we need to get rid of the square root in the denominator. We can do this by multiplying the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{2x+5}-2$ .
Multiply Numerator: Multiply the numerator by the conjugate of the denominator. $\newline$ $(5\sqrt{2x+5}-1) \times (\sqrt{2x+5}-2)$
Apply Distributive Property: Apply the distributive property (FOIL method) to the numerator. $\newline$ $(5\sqrt{2x+5}-1) \times (\sqrt{2x+5}-2) = 5(2x+5) - 10\sqrt{2x+5} + \sqrt{2x+5} - 2$
Combine Like Terms: Combine like terms in the numerator. $\newline$ $5(2x+5) - 10\sqrt{2x+5} + \sqrt{2x+5} - 2 = 10x + 25 - 9\sqrt{2x+5} - 2$
Simplify Numerator: Simplify the terms in the numerator. $\newline$ $10x + 25 - 9\sqrt{2x+5} - 2 = 10x + 23 - 9\sqrt{2x+5}$
Multiply Denominator: Multiply the denominator by the conjugate of the denominator. $\newline$ $(\sqrt{2x+5}+2) \times (\sqrt{2x+5}-2)$
Apply Difference of Squares: Apply the difference of squares formula to the denominator. $\newline$ $(\sqrt{2x+5}+2) \times (\sqrt{2x+5}-2) = (\sqrt{2x+5})^2 - (2)^2$
Simplify Denominator: Simplify the denominator. $(\sqrt{2x+5})^2 - (2)^2 = 2x+5 - 4$
Write Simplified Expression: Combine like terms in the denominator. $\newline$ $2x+5 - 4 = 2x + 1$
Write Simplified Expression: Combine like terms in the denominator. $\newline$ $2x+5 - 4 = 2x + 1$ Write the simplified expression. $\newline$ The simplified expression is $(10x + 23 - 9\sqrt{2x+5}) / (2x + 1)$ .