Solve (-4x-2)6=sqrt{5x-1}

$$(-4x-2)6 = \sqrt{ 5x-1 }$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{8039}i-571}{1152}\approx -0.495659722-0.07783027i$$

Solution Steps

Square both sides of the equation.

$$\left(\left(-4x-2\right)\times 6\right)^{2}=\left(\sqrt{5x-1}\right)^{2}$$

Use the distributive property to multiply -4x-2 by 6.

$$\left(-24x-12\right)^{2}=\left(\sqrt{5x-1}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-24x-12\right)^{2}.

$$576x^{2}+576x+144=\left(\sqrt{5x-1}\right)^{2}$$

Calculate \sqrt{5x-1} to the power of 2 and get 5x-1.

$$576x^{2}+576x+144=5x-1$$

Subtract 5x from both sides.

$$576x^{2}+576x+144-5x=-1$$

Combine 576x and -5x to get 571x.

$$576x^{2}+571x+144=-1$$

Add 1 to both sides.

$$576x^{2}+571x+144+1=0$$

Add 144 and 1 to get 145.

$$576x^{2}+571x+145=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 576 for a, 571 for b, and 145 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-571±\sqrt{571^{2}-4\times 576\times 145}}{2\times 576}$$

Square 571.

$$x=\frac{-571±\sqrt{326041-4\times 576\times 145}}{2\times 576}$$

Multiply -4 times 576.

$$x=\frac{-571±\sqrt{326041-2304\times 145}}{2\times 576}$$

Multiply -2304 times 145.

$$x=\frac{-571±\sqrt{326041-334080}}{2\times 576}$$

Add 326041 to -334080.

$$x=\frac{-571±\sqrt{-8039}}{2\times 576}$$

Take the square root of -8039.

$$x=\frac{-571±\sqrt{8039}i}{2\times 576}$$

Multiply 2 times 576.

$$x=\frac{-571±\sqrt{8039}i}{1152}$$

Now solve the equation x=\frac{-571±\sqrt{8039}i}{1152} when ± is plus. Add -571 to i\sqrt{8039}.

$$x=\frac{-571+\sqrt{8039}i}{1152}$$

Now solve the equation x=\frac{-571±\sqrt{8039}i}{1152} when ± is minus. Subtract i\sqrt{8039} from -571.

$$x=\frac{-\sqrt{8039}i-571}{1152}$$

The equation is now solved.

$$x=\frac{-571+\sqrt{8039}i}{1152} x=\frac{-\sqrt{8039}i-571}{1152}$$

Substitute \frac{-571+\sqrt{8039}i}{1152} for x in the equation \left(-4x-2\right)\times 6=\sqrt{5x-1}.

$$\left(-4\times \frac{-571+\sqrt{8039}i}{1152}-2\right)\times 6=\sqrt{5\times \frac{-571+\sqrt{8039}i}{1152}-1}$$

Simplify. The value x=\frac{-571+\sqrt{8039}i}{1152} does not satisfy the equation.

$$-\frac{5}{48}-\frac{1}{48}i\times 8039^{\frac{1}{2}}=\frac{5}{48}+\frac{1}{48}i\times 8039^{\frac{1}{2}}$$

Substitute \frac{-\sqrt{8039}i-571}{1152} for x in the equation \left(-4x-2\right)\times 6=\sqrt{5x-1}.

$$\left(-4\times \frac{-\sqrt{8039}i-571}{1152}-2\right)\times 6=\sqrt{5\times \frac{-\sqrt{8039}i-571}{1152}-1}$$

Simplify. The value x=\frac{-\sqrt{8039}i-571}{1152} satisfies the equation.

$$\frac{1}{48}i\times 8039^{\frac{1}{2}}-\frac{5}{48}=-\left(\frac{5}{48}-\frac{1}{48}i\times 8039^{\frac{1}{2}}\right)$$

Equation 6\left(-4x-2\right)=\sqrt{5x-1} has a unique solution.

$$x=\frac{-\sqrt{8039}i-571}{1152}$$