Solve 162k^2-108k-18/3sqrt{2}

$$\frac { 162 k ^ { 2 } - 108 k - 18 } { 3 \sqrt { 2 } }$$

Evaluate

$$3\sqrt{2}\left(9k^{2}-6k-1\right)$$

Short Solution Steps

Factor the expressions that are not already factored.

$$\frac{9\times 18\left(k-\left(-\frac{1}{3}\sqrt{2}+\frac{1}{3}\right)\right)\left(k-\left(\frac{1}{3}\sqrt{2}+\frac{1}{3}\right)\right)}{3\sqrt{2}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{3\times 18\left(k-\left(-\frac{1}{3}\sqrt{2}+\frac{1}{3}\right)\right)\left(k-\left(\frac{1}{3}\sqrt{2}+\frac{1}{3}\right)\right)}{\sqrt{2}}$$

Expand the expression.

$$\frac{54k^{2}-36k-6}{\sqrt{2}}$$

Rationalize the denominator of \frac{54k^{2}-36k-6}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\left(54k^{2}-36k-6\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(54k^{2}-36k-6\right)\sqrt{2}}{2}$$

Use the distributive property to multiply 54k^{2}-36k-6 by \sqrt{2}.

$$\frac{54k^{2}\sqrt{2}-36k\sqrt{2}-6\sqrt{2}}{2}$$

Factor

$$27\sqrt{2}\left(k-\frac{1-\sqrt{2}}{3}\right)\left(k-\frac{\sqrt{2}+1}{3}\right)$$