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

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

Evaluate

$\sqrt{2}-1\approx 0.414213562$

Solution Steps

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

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

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

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

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

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

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

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

Anything divided by one gives itself.

$$\sqrt{2}-1$$

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