Solve sqrt{4578}x^2-sqrt{4677521}x+31478=10523

$$\sqrt{ 4578 } { x }^{ 2 } - \sqrt{ 4677521 } x+31478=10523$$

Solve for x (complex solution)

$$x=\frac{\sqrt{21413691138}+i\sqrt{383727960\sqrt{4578}-21413691138}}{9156}\approx 15.982325934+7.366910212i$$

$$x=\frac{-i\sqrt{383727960\sqrt{4578}-21413691138}+\sqrt{21413691138}}{9156}\approx 15.982325934-7.366910212i$$

Steps Using the Quadratic Formula

Subtract 10523 from both sides.

$$\sqrt{4578}x^{2}-\sqrt{4677521}x+31478-10523=0$$

Subtract 10523 from 31478 to get 20955.

$$\sqrt{4578}x^{2}-\sqrt{4677521}x+20955=0$$

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.

$$\sqrt{4578}x^{2}+\left(-\sqrt{4677521}\right)x+20955=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \sqrt{4578} for a, -\sqrt{4677521} for b, and 20955 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\sqrt{4677521}\right)±\sqrt{\left(-\sqrt{4677521}\right)^{2}-4\sqrt{4578}\times 20955}}{2\sqrt{4578}}$$

Square -\sqrt{4677521}.

$$x=\frac{-\left(-\sqrt{4677521}\right)±\sqrt{4677521-4\sqrt{4578}\times 20955}}{2\sqrt{4578}}$$

Multiply -4 times \sqrt{4578}.

$$x=\frac{-\left(-\sqrt{4677521}\right)±\sqrt{4677521+\left(-4\sqrt{4578}\right)\times 20955}}{2\sqrt{4578}}$$

Multiply -4\sqrt{4578} times 20955.

$$x=\frac{-\left(-\sqrt{4677521}\right)±\sqrt{4677521-83820\sqrt{4578}}}{2\sqrt{4578}}$$

Take the square root of 4677521-83820\sqrt{4578}.

$$x=\frac{-\left(-\sqrt{4677521}\right)±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}}$$

The opposite of -\sqrt{4677521} is \sqrt{4677521}.

$$x=\frac{\sqrt{4677521}±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}}$$

Now solve the equation x=\frac{\sqrt{4677521}±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}} when ± is plus. Add \sqrt{4677521} to i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}.

$$x=\frac{\sqrt{4677521}+i\sqrt{83820\sqrt{4578}-4677521}}{2\sqrt{4578}}$$

Divide \sqrt{4677521}+i\sqrt{-4677521+83820\sqrt{4578}} by 2\sqrt{4578}.

$$x=\frac{\sqrt{4578}\left(\sqrt{4677521}+i\sqrt{83820\sqrt{4578}-4677521}\right)}{9156}$$

Now solve the equation x=\frac{\sqrt{4677521}±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}} when ± is minus. Subtract i\sqrt{-\left(4677521-83820\sqrt{4578}\right)} from \sqrt{4677521}.

$$x=\frac{-i\sqrt{83820\sqrt{4578}-4677521}+\sqrt{4677521}}{2\sqrt{4578}}$$

Divide \sqrt{4677521}-i\sqrt{-4677521+83820\sqrt{4578}} by 2\sqrt{4578}.

$$x=\frac{\sqrt{4578}\left(-i\sqrt{83820\sqrt{4578}-4677521}+\sqrt{4677521}\right)}{9156}$$

The equation is now solved.

$$x=\frac{\sqrt{4578}\left(\sqrt{4677521}+i\sqrt{83820\sqrt{4578}-4677521}\right)}{9156} x=\frac{\sqrt{4578}\left(-i\sqrt{83820\sqrt{4578}-4677521}+\sqrt{4677521}\right)}{9156}$$

Steps for Completing the Square

Subtract 31478 from both sides.

$$\sqrt{4578}x^{2}-\sqrt{4677521}x=10523-31478$$

Subtract 31478 from 10523 to get -20955.

$$\sqrt{4578}x^{2}-\sqrt{4677521}x=-20955$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$\sqrt{4578}x^{2}+\left(-\sqrt{4677521}\right)x=-20955$$

Divide both sides by \sqrt{4578}.

$$\frac{\sqrt{4578}x^{2}+\left(-\sqrt{4677521}\right)x}{\sqrt{4578}}=-\frac{20955}{\sqrt{4578}}$$

Dividing by \sqrt{4578} undoes the multiplication by \sqrt{4578}.

$$x^{2}+\left(-\frac{\sqrt{4677521}}{\sqrt{4578}}\right)x=-\frac{20955}{\sqrt{4578}}$$

Divide -\sqrt{4677521} by \sqrt{4578}.

$$x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x=-\frac{20955}{\sqrt{4578}}$$

Divide -20955 by \sqrt{4578}.

$$x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x=-\frac{6985\sqrt{4578}}{1526}$$

Divide -\frac{\sqrt{21413691138}}{4578}, the coefficient of the x term, by 2 to get -\frac{\sqrt{21413691138}}{9156}. Then add the square of -\frac{\sqrt{21413691138}}{9156} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x+\left(-\frac{\sqrt{21413691138}}{9156}\right)^{2}=-\frac{6985\sqrt{4578}}{1526}+\left(-\frac{\sqrt{21413691138}}{9156}\right)^{2}$$

Square -\frac{\sqrt{21413691138}}{9156}.

$$x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x+\frac{4677521}{18312}=-\frac{6985\sqrt{4578}}{1526}+\frac{4677521}{18312}$$

Factor x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x+\frac{4677521}{18312}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{\sqrt{21413691138}}{9156}\right)^{2}=-\frac{6985\sqrt{4578}}{1526}+\frac{4677521}{18312}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{\sqrt{21413691138}}{9156}\right)^{2}}=\sqrt{-\frac{6985\sqrt{4578}}{1526}+\frac{4677521}{18312}}$$

Simplify.

$$x-\frac{\sqrt{21413691138}}{9156}=\frac{i\sqrt{383727960\sqrt{4578}-21413691138}}{9156} x-\frac{\sqrt{21413691138}}{9156}=-\frac{i\sqrt{383727960\sqrt{4578}-21413691138}}{9156}$$

Add \frac{\sqrt{21413691138}}{9156} to both sides of the equation.

$$x=\frac{\sqrt{21413691138}+i\sqrt{383727960\sqrt{4578}-21413691138}}{9156} x=\frac{-i\sqrt{383727960\sqrt{4578}-21413691138}+\sqrt{21413691138}}{9156}$$