Solve sqrt{7x}-82x=2

$$\sqrt{ 7x } -82x=2$$

Solve for x (complex solution)

$$x=\frac{-321+\sqrt{4543}i}{13448}\approx -0.02386972+0.00501203i$$

Solution Steps

Subtract -82x from both sides of the equation.

$$\sqrt{7x}=2+82x$$

Square both sides of the equation.

$$\left(\sqrt{7x}\right)^{2}=\left(2+82x\right)^{2}$$

Calculate \sqrt{7x} to the power of 2 and get 7x.

$$7x=\left(2+82x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+82x\right)^{2}.

$$7x=4+328x+6724x^{2}$$

Subtract 4 from both sides.

$$7x-4=328x+6724x^{2}$$

Subtract 328x from both sides.

$$7x-4-328x=6724x^{2}$$

Combine 7x and -328x to get -321x.

$$-321x-4=6724x^{2}$$

Subtract 6724x^{2} from both sides.

$$-321x-4-6724x^{2}=0$$

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.

$$-6724x^{2}-321x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6724 for a, -321 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-321\right)±\sqrt{\left(-321\right)^{2}-4\left(-6724\right)\left(-4\right)}}{2\left(-6724\right)}$$

Square -321.

$$x=\frac{-\left(-321\right)±\sqrt{103041-4\left(-6724\right)\left(-4\right)}}{2\left(-6724\right)}$$

Multiply -4 times -6724.

$$x=\frac{-\left(-321\right)±\sqrt{103041+26896\left(-4\right)}}{2\left(-6724\right)}$$

Multiply 26896 times -4.

$$x=\frac{-\left(-321\right)±\sqrt{103041-107584}}{2\left(-6724\right)}$$

Add 103041 to -107584.

$$x=\frac{-\left(-321\right)±\sqrt{-4543}}{2\left(-6724\right)}$$

Take the square root of -4543.

$$x=\frac{-\left(-321\right)±\sqrt{4543}i}{2\left(-6724\right)}$$

The opposite of -321 is 321.

$$x=\frac{321±\sqrt{4543}i}{2\left(-6724\right)}$$

Multiply 2 times -6724.

$$x=\frac{321±\sqrt{4543}i}{-13448}$$

Now solve the equation x=\frac{321±\sqrt{4543}i}{-13448} when ± is plus. Add 321 to i\sqrt{4543}.

$$x=\frac{321+\sqrt{4543}i}{-13448}$$

Divide 321+i\sqrt{4543} by -13448.

$$x=\frac{-\sqrt{4543}i-321}{13448}$$

Now solve the equation x=\frac{321±\sqrt{4543}i}{-13448} when ± is minus. Subtract i\sqrt{4543} from 321.

$$x=\frac{-\sqrt{4543}i+321}{-13448}$$

Divide 321-i\sqrt{4543} by -13448.

$$x=\frac{-321+\sqrt{4543}i}{13448}$$

The equation is now solved.

$$x=\frac{-\sqrt{4543}i-321}{13448} x=\frac{-321+\sqrt{4543}i}{13448}$$

Substitute \frac{-\sqrt{4543}i-321}{13448} for x in the equation \sqrt{7x}-82x=2.

$$\sqrt{7\times \frac{-\sqrt{4543}i-321}{13448}}-82\times \frac{-\sqrt{4543}i-321}{13448}=2$$

Simplify. The value x=\frac{-\sqrt{4543}i-321}{13448} does not satisfy the equation.

$$\frac{157}{82}+\frac{1}{82}i\times 4543^{\frac{1}{2}}=2$$

Substitute \frac{-321+\sqrt{4543}i}{13448} for x in the equation \sqrt{7x}-82x=2.

$$\sqrt{7\times \frac{-321+\sqrt{4543}i}{13448}}-82\times \frac{-321+\sqrt{4543}i}{13448}=2$$

Simplify. The value x=\frac{-321+\sqrt{4543}i}{13448} satisfies the equation.

$$2=2$$

Equation \sqrt{7x}=82x+2 has a unique solution.

$$x=\frac{-321+\sqrt{4543}i}{13448}$$