How to solve 6/(6 - sqrt(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 6/(6 - sqrt(5)) | Equatix

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Question

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

Evaluate

$\frac{6\left(\sqrt{5}+6\right)}{31}\approx 1.594077673$

Solution Steps

Rationalize the denominator of $\frac{6}{6-\sqrt{5}}$ by multiplying numerator and denominator by $6+\sqrt{5}$.

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

Consider $\left(6-\sqrt{5}\right)\left(6+\sqrt{5}\right)$. Multiplication can be transformed into difference of squares using the rule: $\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$.

$$\frac{6\left(6+\sqrt{5}\right)}{6^{2}-\left(\sqrt{5}\right)^{2}}$$

Square $6$. Square $\sqrt{5}$.

$$\frac{6\left(6+\sqrt{5}\right)}{36-5}$$

Subtract $5$ from $36$ to get $31$.

$$\frac{6\left(6+\sqrt{5}\right)}{31}$$

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

$$\frac{36+6\sqrt{5}}{31}$$

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