How to solve 10/(sqrt(13) - 5)

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.

Solve 10/(sqrt(13) - 5) | Equatix

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Question

$$\frac{10}{\sqrt{13}-5}$$

Evaluate

$\frac{-5\sqrt{13}-25}{6}\approx -7.17129273$

Solution Steps

Rationalize the denominator of $\frac{10}{\sqrt{13}-5}$ by multiplying numerator and denominator by $\sqrt{13}+5$.

$$\frac{10\left(\sqrt{13}+5\right)}{\left(\sqrt{13}-5\right)\left(\sqrt{13}+5\right)}$$

Consider $\left(\sqrt{13}-5\right)\left(\sqrt{13}+5\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{10\left(\sqrt{13}+5\right)}{\left(\sqrt{13}\right)^{2}-5^{2}}$$

Square $\sqrt{13}$. Square $5$.

$$\frac{10\left(\sqrt{13}+5\right)}{13-25}$$

Subtract $25$ from $13$ to get $-12$.

$$\frac{10\left(\sqrt{13}+5\right)}{-12}$$

Divide $10\left(\sqrt{13}+5\right)$ by $-12$ to get $-\frac{5}{6}\left(\sqrt{13}+5\right)$.

$$-\frac{5}{6}\left(\sqrt{13}+5\right)$$

Use the distributive property to multiply $-\frac{5}{6}$ by $\sqrt{13}+5$.

$$-\frac{5}{6}\sqrt{13}-\frac{5}{6}\times 5$$

Express $-\frac{5}{6}\times 5$ as a single fraction.

$$-\frac{5}{6}\sqrt{13}+\frac{-5\times 5}{6}$$

Multiply $-5$ and $5$ to get $-25$.

$$-\frac{5}{6}\sqrt{13}+\frac{-25}{6}$$

Fraction $\frac{-25}{6}$ can be rewritten as $-\frac{25}{6}$ by extracting the negative sign.

$$-\frac{5}{6}\sqrt{13}-\frac{25}{6}$$

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