1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

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Question

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

Evaluate

$\text{Indeterminate}$

Solution Steps

Rationalize the denominator of $\frac{5}{1+\sqrt{-5}}$ by multiplying numerator and denominator by $1-\sqrt{-5}$.

$$\frac{5\left(1-\sqrt{-5}\right)}{\left(1+\sqrt{-5}\right)\left(1-\sqrt{-5}\right)}$$

Consider $\left(1+\sqrt{-5}\right)\left(1-\sqrt{-5}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{5\left(1-\sqrt{-5}\right)}{1^{2}-\left(\sqrt{-5}\right)^{2}}$$

Square $1$. Square $\sqrt{-5}$.

$$\frac{5\left(1-\sqrt{-5}\right)}{1+5}$$

Subtract $-5$ from $1$ to get $6$.

$$\frac{5\left(1-\sqrt{-5}\right)}{6}$$

Use the distributive property to multiply $5$ by $1-\sqrt{-5}$.

$$\frac{5-5\sqrt{-5}}{6}$$

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