Solve -36=30x-2(sqrt{27x}*sqrt{3x})

$$- 36 = 30 x - 2 ( \sqrt { 27 x } \cdot \sqrt { 3 x } )$$

Solve for x (complex solution)

$$x=-3$$

Solution Steps

Subtract 30x from both sides of the equation.

$$-36-30x=-2\sqrt{27x}\sqrt{3x}$$

Square both sides of the equation.

$$\left(-36-30x\right)^{2}=\left(-2\sqrt{27x}\sqrt{3x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-36-30x\right)^{2}.

$$1296+2160x+900x^{2}=\left(-2\sqrt{27x}\sqrt{3x}\right)^{2}$$

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

$$1296+2160x+900x^{2}=\left(-2\right)^{2}\left(\sqrt{27x}\right)^{2}\left(\sqrt{3x}\right)^{2}$$

Calculate -2 to the power of 2 and get 4.

$$1296+2160x+900x^{2}=4\left(\sqrt{27x}\right)^{2}\left(\sqrt{3x}\right)^{2}$$

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

$$1296+2160x+900x^{2}=4\times 27x\left(\sqrt{3x}\right)^{2}$$

Multiply 4 and 27 to get 108.

$$1296+2160x+900x^{2}=108x\left(\sqrt{3x}\right)^{2}$$

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

$$1296+2160x+900x^{2}=108x\times 3x$$

Multiply 108 and 3 to get 324.

$$1296+2160x+900x^{2}=324xx$$

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

$$1296+2160x+900x^{2}=324x^{2}$$

Subtract 324x^{2} from both sides.

$$1296+2160x+900x^{2}-324x^{2}=0$$

Combine 900x^{2} and -324x^{2} to get 576x^{2}.

$$1296+2160x+576x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$576x^{2}+2160x+1296=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 576 for a, 2160 for b, and 1296 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2160±\sqrt{2160^{2}-4\times 576\times 1296}}{2\times 576}$$

Square 2160.

$$x=\frac{-2160±\sqrt{4665600-4\times 576\times 1296}}{2\times 576}$$

Multiply -4 times 576.

$$x=\frac{-2160±\sqrt{4665600-2304\times 1296}}{2\times 576}$$

Multiply -2304 times 1296.

$$x=\frac{-2160±\sqrt{4665600-2985984}}{2\times 576}$$

Add 4665600 to -2985984.

$$x=\frac{-2160±\sqrt{1679616}}{2\times 576}$$

Take the square root of 1679616.

$$x=\frac{-2160±1296}{2\times 576}$$

Multiply 2 times 576.

$$x=\frac{-2160±1296}{1152}$$

Now solve the equation x=\frac{-2160±1296}{1152} when ± is plus. Add -2160 to 1296.

$$x=-\frac{864}{1152}$$

Reduce the fraction \frac{-864}{1152} to lowest terms by extracting and canceling out 288.

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

Now solve the equation x=\frac{-2160±1296}{1152} when ± is minus. Subtract 1296 from -2160.

$$x=-\frac{3456}{1152}$$

Divide -3456 by 1152.

$$x=-3$$

The equation is now solved.

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

Substitute -\frac{3}{4} for x in the equation -36=30x-2\sqrt{27x}\sqrt{3x}.

$$-36=30\left(-\frac{3}{4}\right)-2\sqrt{27\left(-\frac{3}{4}\right)}\sqrt{3\left(-\frac{3}{4}\right)}$$

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

$$-36=-9$$

Substitute -3 for x in the equation -36=30x-2\sqrt{27x}\sqrt{3x}.

$$-36=30\left(-3\right)-2\sqrt{27\left(-3\right)}\sqrt{3\left(-3\right)}$$

Simplify. The value x=-3 satisfies the equation.

$$-36=-36$$

Equation -30x-36=-2\sqrt{3x}\sqrt{27x} has a unique solution.

$$x=-3$$