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Simplify. Rationalize the denominator. $\newline$ $\frac{2}{-2 + \sqrt{5}}$

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Simplify radical expressions using conjugates

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Q. Simplify. Rationalize the denominator.

\newline

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

Find Conjugate: Select the conjugate of $-2 + \sqrt{5}$ . $\newline$ The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$ , so the conjugate of $-2 + \sqrt{5}$ is $-2 - \sqrt{5}$ .
Multiply by Conjugate: Multiply the numerator and the denominator by the conjugate of the denominator. $\newline$ To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator: $\newline$ $(2 \cdot (-2 - \sqrt{5})) / ((-2 + \sqrt{5}) \cdot (-2 - \sqrt{5}))$
Simplify Numerator: Simplify the numerator. $\newline$ Now we distribute the $2$ in the numerator across the conjugate: $\newline$ $2 \times (-2) + 2 \times (-\sqrt{5})$ $\newline$ $= -4 - 2\sqrt{5}$
Simplify Denominator: Simplify the denominator. $\newline$ We use the difference of squares formula, which states that $(a + b)(a - b) = a^2 - b^2$ : $\newline$ $(-2)^2 - (\sqrt{5})^2$ $\newline$ = $4$ - $5$ $\newline$ = $-1$
Combine Numerator and Denominator: Combine the simplified numerator and denominator. $\newline$ Now we have: $\newline$ $(-4 - 2\sqrt{5}) / (-1)$ $\newline$ When we divide by $-1$ , we change the sign of each term in the numerator: $\newline$ $4 + 2\sqrt{5}$

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