Solve 473^-4=x^2/64-x

$${ 473 }^{ -4 } = \frac{ { x }^{ 2 } }{ 64-x }$$

Solve for x

$$x=\frac{\sqrt{12813994352897}-1}{100109330882}\approx 0.000035758$$

$$x=\frac{-\sqrt{12813994352897}-1}{100109330882}\approx -0.000035758$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 64 since division by zero is not defined. Multiply both sides of the equation by -x+64.

$$\left(-x+64\right)\times 473^{-4}=x^{2}$$

Calculate 473 to the power of -4 and get \frac{1}{50054665441}.

$$\left(-x+64\right)\times \frac{1}{50054665441}=x^{2}$$

Use the distributive property to multiply -x+64 by \frac{1}{50054665441}.

$$-\frac{1}{50054665441}x+\frac{64}{50054665441}=x^{2}$$

Subtract x^{2} from both sides.

$$-\frac{1}{50054665441}x+\frac{64}{50054665441}-x^{2}=0$$

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.

$$-x^{2}-\frac{1}{50054665441}x+\frac{64}{50054665441}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -\frac{1}{50054665441} for b, and \frac{64}{50054665441} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{1}{50054665441}\right)±\sqrt{\left(-\frac{1}{50054665441}\right)^{2}-4\left(-1\right)\times \frac{64}{50054665441}}}{2\left(-1\right)}$$

Square -\frac{1}{50054665441} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{1}{50054665441}\right)±\sqrt{\frac{1}{2505469532410439724481}-4\left(-1\right)\times \frac{64}{50054665441}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-\frac{1}{50054665441}\right)±\sqrt{\frac{1}{2505469532410439724481}+4\times \frac{64}{50054665441}}}{2\left(-1\right)}$$

Multiply 4 times \frac{64}{50054665441}.

$$x=\frac{-\left(-\frac{1}{50054665441}\right)±\sqrt{\frac{1}{2505469532410439724481}+\frac{256}{50054665441}}}{2\left(-1\right)}$$

Add \frac{1}{2505469532410439724481} to \frac{256}{50054665441} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{1}{50054665441}\right)±\sqrt{\frac{12813994352897}{2505469532410439724481}}}{2\left(-1\right)}$$

Take the square root of \frac{12813994352897}{2505469532410439724481}.

$$x=\frac{-\left(-\frac{1}{50054665441}\right)±\frac{\sqrt{12813994352897}}{50054665441}}{2\left(-1\right)}$$

The opposite of -\frac{1}{50054665441} is \frac{1}{50054665441}.

$$x=\frac{\frac{1}{50054665441}±\frac{\sqrt{12813994352897}}{50054665441}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{\frac{1}{50054665441}±\frac{\sqrt{12813994352897}}{50054665441}}{-2}$$

Now solve the equation x=\frac{\frac{1}{50054665441}±\frac{\sqrt{12813994352897}}{50054665441}}{-2} when ± is plus. Add \frac{1}{50054665441} to \frac{\sqrt{12813994352897}}{50054665441}.

$$x=\frac{\sqrt{12813994352897}+1}{-2\times 50054665441}$$

Divide \frac{1+\sqrt{12813994352897}}{50054665441} by -2.

$$x=\frac{-\sqrt{12813994352897}-1}{100109330882}$$

Now solve the equation x=\frac{\frac{1}{50054665441}±\frac{\sqrt{12813994352897}}{50054665441}}{-2} when ± is minus. Subtract \frac{\sqrt{12813994352897}}{50054665441} from \frac{1}{50054665441}.

$$x=\frac{1-\sqrt{12813994352897}}{-2\times 50054665441}$$

Divide \frac{1-\sqrt{12813994352897}}{50054665441} by -2.

$$x=\frac{\sqrt{12813994352897}-1}{100109330882}$$

The equation is now solved.

$$x=\frac{-\sqrt{12813994352897}-1}{100109330882} x=\frac{\sqrt{12813994352897}-1}{100109330882}$$

Steps for Completing the Square

Variable x cannot be equal to 64 since division by zero is not defined. Multiply both sides of the equation by -x+64.

$$\left(-x+64\right)\times 473^{-4}=x^{2}$$

Calculate 473 to the power of -4 and get \frac{1}{50054665441}.

$$\left(-x+64\right)\times \frac{1}{50054665441}=x^{2}$$

Use the distributive property to multiply -x+64 by \frac{1}{50054665441}.

$$-\frac{1}{50054665441}x+\frac{64}{50054665441}=x^{2}$$

Subtract x^{2} from both sides.

$$-\frac{1}{50054665441}x+\frac{64}{50054665441}-x^{2}=0$$

Subtract \frac{64}{50054665441} from both sides. Anything subtracted from zero gives its negation.

$$-\frac{1}{50054665441}x-x^{2}=-\frac{64}{50054665441}$$

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.

$$-x^{2}-\frac{1}{50054665441}x=-\frac{64}{50054665441}$$

Divide both sides by -1.

$$\frac{-x^{2}-\frac{1}{50054665441}x}{-1}=-\frac{\frac{64}{50054665441}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{\frac{1}{50054665441}}{-1}\right)x=-\frac{\frac{64}{50054665441}}{-1}$$

Divide -\frac{1}{50054665441} by -1.

$$x^{2}+\frac{1}{50054665441}x=-\frac{\frac{64}{50054665441}}{-1}$$

Divide -\frac{64}{50054665441} by -1.

$$x^{2}+\frac{1}{50054665441}x=\frac{64}{50054665441}$$

Divide \frac{1}{50054665441}, the coefficient of the x term, by 2 to get \frac{1}{100109330882}. Then add the square of \frac{1}{100109330882} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{50054665441}x+\left(\frac{1}{100109330882}\right)^{2}=\frac{64}{50054665441}+\left(\frac{1}{100109330882}\right)^{2}$$

Square \frac{1}{100109330882} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{50054665441}x+\frac{1}{10021878129641758897924}=\frac{64}{50054665441}+\frac{1}{10021878129641758897924}$$

Add \frac{64}{50054665441} to \frac{1}{10021878129641758897924} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{50054665441}x+\frac{1}{10021878129641758897924}=\frac{12813994352897}{10021878129641758897924}$$

Factor x^{2}+\frac{1}{50054665441}x+\frac{1}{10021878129641758897924}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{100109330882}\right)^{2}=\frac{12813994352897}{10021878129641758897924}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{100109330882}\right)^{2}}=\sqrt{\frac{12813994352897}{10021878129641758897924}}$$

Simplify.

$$x+\frac{1}{100109330882}=\frac{\sqrt{12813994352897}}{100109330882} x+\frac{1}{100109330882}=-\frac{\sqrt{12813994352897}}{100109330882}$$

Subtract \frac{1}{100109330882} from both sides of the equation.

$$x=\frac{\sqrt{12813994352897}-1}{100109330882} x=\frac{-\sqrt{12813994352897}-1}{100109330882}$$