Solve 18=15x-(sqrt{27x}*sqrt{3x})

$$18 = 15 x - ( \sqrt { 27 x } \cdot \sqrt { 3 x } )$$

Solve for x

$$x=3$$

Solution Steps

Subtract 15x from both sides of the equation.

$$18-15x=-\sqrt{27x}\sqrt{3x}$$

Reorder the terms.

$$-15x+18=-\sqrt{3x}\sqrt{27x}$$

Square both sides of the equation.

$$\left(-15x+18\right)^{2}=\left(-\sqrt{3x}\sqrt{27x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-15x+18\right)^{2}.

$$225x^{2}-540x+324=\left(-\sqrt{3x}\sqrt{27x}\right)^{2}$$

Expand \left(-\sqrt{3x}\sqrt{27x}\right)^{2}.

$$225x^{2}-540x+324=\left(-1\right)^{2}\left(\sqrt{3x}\right)^{2}\left(\sqrt{27x}\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$225x^{2}-540x+324=1\left(\sqrt{3x}\right)^{2}\left(\sqrt{27x}\right)^{2}$$

Calculate \sqrt{3x} to the power of 2 and get 3x.

$$225x^{2}-540x+324=1\times 3x\left(\sqrt{27x}\right)^{2}$$

Multiply 1 and 3 to get 3.

$$225x^{2}-540x+324=3x\left(\sqrt{27x}\right)^{2}$$

Calculate \sqrt{27x} to the power of 2 and get 27x.

$$225x^{2}-540x+324=3x\times 27x$$

Multiply 3 and 27 to get 81.

$$225x^{2}-540x+324=81xx$$

Multiply x and x to get x^{2}.

$$225x^{2}-540x+324=81x^{2}$$

Subtract 81x^{2} from both sides.

$$225x^{2}-540x+324-81x^{2}=0$$

Combine 225x^{2} and -81x^{2} to get 144x^{2}.

$$144x^{2}-540x+324=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 144 for a, -540 for b, and 324 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-540\right)±\sqrt{\left(-540\right)^{2}-4\times 144\times 324}}{2\times 144}$$

Square -540.

$$x=\frac{-\left(-540\right)±\sqrt{291600-4\times 144\times 324}}{2\times 144}$$

Multiply -4 times 144.

$$x=\frac{-\left(-540\right)±\sqrt{291600-576\times 324}}{2\times 144}$$

Multiply -576 times 324.

$$x=\frac{-\left(-540\right)±\sqrt{291600-186624}}{2\times 144}$$

Add 291600 to -186624.

$$x=\frac{-\left(-540\right)±\sqrt{104976}}{2\times 144}$$

Take the square root of 104976.

$$x=\frac{-\left(-540\right)±324}{2\times 144}$$

The opposite of -540 is 540.

$$x=\frac{540±324}{2\times 144}$$

Multiply 2 times 144.

$$x=\frac{540±324}{288}$$

Now solve the equation x=\frac{540±324}{288} when ± is plus. Add 540 to 324.

$$x=\frac{864}{288}$$

Divide 864 by 288.

$$x=3$$

Now solve the equation x=\frac{540±324}{288} when ± is minus. Subtract 324 from 540.

$$x=\frac{216}{288}$$

Reduce the fraction \frac{216}{288} to lowest terms by extracting and canceling out 72.

$$x=\frac{3}{4}$$

The equation is now solved.

$$x=3 x=\frac{3}{4}$$

Substitute 3 for x in the equation 18=15x-\sqrt{27x}\sqrt{3x}.

$$18=15\times 3-\sqrt{27\times 3}\sqrt{3\times 3}$$

Simplify. The value x=3 satisfies the equation.

$$18=18$$

Substitute \frac{3}{4} for x in the equation 18=15x-\sqrt{27x}\sqrt{3x}.

$$18=15\times \frac{3}{4}-\sqrt{27\times \frac{3}{4}}\sqrt{3\times \frac{3}{4}}$$

Simplify. The value x=\frac{3}{4} does not satisfy the equation.

$$18=\frac{9}{2}$$

Equation -15x+18=-\sqrt{3x}\sqrt{27x} has a unique solution.

$$x=3$$