Solve x=-frac{12x+sqrt{12x^2-4(5x-1)}}{2}

$$x = - \frac { 12 x + \sqrt { 12 x ^ { 2 } - 4 ( 5 x - 1 ) } } { 2 }$$

Solve for x

$$x=\frac{-\sqrt{209}-5}{92}\approx -0.211487308$$

Solution Steps

Use the distributive property to multiply -4 by 5x-1.

$$x=-\frac{12x+\sqrt{12x^{2}-20x+4}}{2}$$

Add \frac{12x+\sqrt{12x^{2}-20x+4}}{2} to both sides.

$$x+\frac{12x+\sqrt{12x^{2}-20x+4}}{2}=0$$

Multiply both sides of the equation by 2.

$$2x+12x+\sqrt{12x^{2}-20x+4}=0$$

Reorder the terms.

$$\sqrt{12x^{2}-20x+4}+2x+12x=0$$

Combine 2x and 12x to get 14x.

$$\sqrt{12x^{2}-20x+4}+14x=0$$

Subtract 14x from both sides of the equation.

$$\sqrt{12x^{2}-20x+4}=-14x$$

Square both sides of the equation.

$$\left(\sqrt{12x^{2}-20x+4}\right)^{2}=\left(-14x\right)^{2}$$

Calculate \sqrt{12x^{2}-20x+4} to the power of 2 and get 12x^{2}-20x+4.

$$12x^{2}-20x+4=\left(-14x\right)^{2}$$

Expand \left(-14x\right)^{2}.

$$12x^{2}-20x+4=\left(-14\right)^{2}x^{2}$$

Calculate -14 to the power of 2 and get 196.

$$12x^{2}-20x+4=196x^{2}$$

Subtract 196x^{2} from both sides.

$$12x^{2}-20x+4-196x^{2}=0$$

Combine 12x^{2} and -196x^{2} to get -184x^{2}.

$$-184x^{2}-20x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -184 for a, -20 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\left(-184\right)\times 4}}{2\left(-184\right)}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\left(-184\right)\times 4}}{2\left(-184\right)}$$

Multiply -4 times -184.

$$x=\frac{-\left(-20\right)±\sqrt{400+736\times 4}}{2\left(-184\right)}$$

Multiply 736 times 4.

$$x=\frac{-\left(-20\right)±\sqrt{400+2944}}{2\left(-184\right)}$$

Add 400 to 2944.

$$x=\frac{-\left(-20\right)±\sqrt{3344}}{2\left(-184\right)}$$

Take the square root of 3344.

$$x=\frac{-\left(-20\right)±4\sqrt{209}}{2\left(-184\right)}$$

The opposite of -20 is 20.

$$x=\frac{20±4\sqrt{209}}{2\left(-184\right)}$$

Multiply 2 times -184.

$$x=\frac{20±4\sqrt{209}}{-368}$$

Now solve the equation x=\frac{20±4\sqrt{209}}{-368} when ± is plus. Add 20 to 4\sqrt{209}.

$$x=\frac{4\sqrt{209}+20}{-368}$$

Divide 20+4\sqrt{209} by -368.

$$x=\frac{-\sqrt{209}-5}{92}$$

Now solve the equation x=\frac{20±4\sqrt{209}}{-368} when ± is minus. Subtract 4\sqrt{209} from 20.

$$x=\frac{20-4\sqrt{209}}{-368}$$

Divide 20-4\sqrt{209} by -368.

$$x=\frac{\sqrt{209}-5}{92}$$

The equation is now solved.

$$x=\frac{-\sqrt{209}-5}{92} x=\frac{\sqrt{209}-5}{92}$$

Substitute \frac{-\sqrt{209}-5}{92} for x in the equation x=-\frac{12x+\sqrt{12x^{2}-4\left(5x-1\right)}}{2}.

$$\frac{-\sqrt{209}-5}{92}=-\frac{12\times \frac{-\sqrt{209}-5}{92}+\sqrt{12\times \left(\frac{-\sqrt{209}-5}{92}\right)^{2}-4\left(5\times \frac{-\sqrt{209}-5}{92}-1\right)}}{2}$$

Simplify. The value x=\frac{-\sqrt{209}-5}{92} satisfies the equation.

$$-\frac{1}{92}\times 209^{\frac{1}{2}}-\frac{5}{92}=-\frac{1}{92}\times 209^{\frac{1}{2}}-\frac{5}{92}$$

Substitute \frac{\sqrt{209}-5}{92} for x in the equation x=-\frac{12x+\sqrt{12x^{2}-4\left(5x-1\right)}}{2}.

$$\frac{\sqrt{209}-5}{92}=-\frac{12\times \frac{\sqrt{209}-5}{92}+\sqrt{12\times \left(\frac{\sqrt{209}-5}{92}\right)^{2}-4\left(5\times \frac{\sqrt{209}-5}{92}-1\right)}}{2}$$

Simplify. The value x=\frac{\sqrt{209}-5}{92} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{1}{92}\times 209^{\frac{1}{2}}-\frac{5}{92}=-\frac{13}{92}\times 209^{\frac{1}{2}}+\frac{65}{92}$$

Equation \sqrt{12x^{2}-20x+4}=-14x has a unique solution.

$$x=\frac{-\sqrt{209}-5}{92}$$