How to solve 2/(1 + sqrt(2))

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 2/(1 + sqrt(2)) | Equatix

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Question

$$\frac{2}{1+\sqrt{2}}$$

Evaluate

$2\sqrt{2}-2\approx 0.828427125$

Solution Steps

Rationalize the denominator of $\frac{2}{1+\sqrt{2}}$ by multiplying numerator and denominator by $1-\sqrt{2}$.

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

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

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

Square $1$. Square $\sqrt{2}$.

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

Subtract $2$ from $1$ to get $-1$.

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

Anything divided by -1 gives its opposite.

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

Use the distributive property to multiply $-2$ by $1-\sqrt{2}$.

$$-2+2\sqrt{2}$$

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