Solve x=frac{12x+sqrt{-12x^2-4(5x)}}{2}

$$x = \frac { 12 x + \sqrt { - 12 x ^ { 2 } - 4 ( 5 x ) } } { 2 }$$

Solve for x

$$x=-\frac{5}{28}\approx -0.178571429$$

$$x=0$$

Solution Steps

Multiply 4 and 5 to get 20.

$$x=\frac{12x+\sqrt{-12x^{2}-20x}}{2}$$

Subtract \frac{12x+\sqrt{-12x^{2}-20x}}{2} from both sides.

$$x-\frac{12x+\sqrt{-12x^{2}-20x}}{2}=0$$

Multiply both sides of the equation by 2.

$$2x-\left(12x+\sqrt{-12x^{2}-20x}\right)=0$$

To find the opposite of 12x+\sqrt{-12x^{2}-20x}, find the opposite of each term.

$$2x-12x-\sqrt{-12x^{2}-20x}=0$$

Combine 2x and -12x to get -10x.

$$-10x-\sqrt{-12x^{2}-20x}=0$$

Subtract -10x from both sides of the equation.

$$-\sqrt{-12x^{2}-20x}=10x$$

Square both sides of the equation.

$$\left(-\sqrt{-12x^{2}-20x}\right)^{2}=\left(10x\right)^{2}$$

Expand \left(-\sqrt{-12x^{2}-20x}\right)^{2}.

$$\left(-1\right)^{2}\left(\sqrt{-12x^{2}-20x}\right)^{2}=\left(10x\right)^{2}$$

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

$$1\left(\sqrt{-12x^{2}-20x}\right)^{2}=\left(10x\right)^{2}$$

Calculate \sqrt{-12x^{2}-20x} to the power of 2 and get -12x^{2}-20x.

$$1\left(-12x^{2}-20x\right)=\left(10x\right)^{2}$$

Use the distributive property to multiply 1 by -12x^{2}-20x.

$$-12x^{2}-20x=\left(10x\right)^{2}$$

Expand \left(10x\right)^{2}.

$$-12x^{2}-20x=10^{2}x^{2}$$

Calculate 10 to the power of 2 and get 100.

$$-12x^{2}-20x=100x^{2}$$

Subtract 100x^{2} from both sides.

$$-12x^{2}-20x-100x^{2}=0$$

Combine -12x^{2} and -100x^{2} to get -112x^{2}.

$$-112x^{2}-20x=0$$

Factor out x.

$$x\left(-112x-20\right)=0$$

To find equation solutions, solve x=0 and -112x-20=0.

$$x=0 x=-\frac{5}{28}$$

Substitute 0 for x in the equation x=\frac{12x+\sqrt{-12x^{2}-4\times 5x}}{2}.

$$0=\frac{12\times 0+\sqrt{-12\times 0^{2}-4\times 5\times 0}}{2}$$

Simplify. The value x=0 satisfies the equation.

$$0=0$$

Substitute -\frac{5}{28} for x in the equation x=\frac{12x+\sqrt{-12x^{2}-4\times 5x}}{2}.

$$-\frac{5}{28}=\frac{12\left(-\frac{5}{28}\right)+\sqrt{-12\left(-\frac{5}{28}\right)^{2}-4\times 5\left(-\frac{5}{28}\right)}}{2}$$

Simplify. The value x=-\frac{5}{28} satisfies the equation.

$$-\frac{5}{28}=-\frac{5}{28}$$

List all solutions of -\sqrt{-12x^{2}-20x}=10x.

$$x=0 x=-\frac{5}{28}$$