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

$$\frac { 2 } { \sqrt { 7 } + \sqrt { 2 } }$$

Evaluate

$\frac{2\left(\sqrt{7}-\sqrt{2}\right)}{5}\approx 0.492615099$

Solution Steps

Rationalize the denominator of $\frac{2}{\sqrt{7}+\sqrt{2}}$ by multiplying numerator and denominator by $\sqrt{7}-\sqrt{2}$.

$$\frac{2\left(\sqrt{7}-\sqrt{2}\right)}{\left(\sqrt{7}+\sqrt{2}\right)\left(\sqrt{7}-\sqrt{2}\right)}$$

Consider $\left(\sqrt{7}+\sqrt{2}\right)\left(\sqrt{7}-\sqrt{2}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{2\left(\sqrt{7}-\sqrt{2}\right)}{\left(\sqrt{7}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square $\sqrt{7}$. Square $\sqrt{2}$.

$$\frac{2\left(\sqrt{7}-\sqrt{2}\right)}{7-2}$$

Subtract $2$ from $7$ to get $5$.

$$\frac{2\left(\sqrt{7}-\sqrt{2}\right)}{5}$$

Use the distributive property to multiply $2$ by $\sqrt{7}-\sqrt{2}$.

$$\frac{2\sqrt{7}-2\sqrt{2}}{5}$$

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