Solve 400=x^2/left(284-xright)^2

$$400= \frac{ { x }^{ 2 } }{ { \left(284-x \right) }^{ 2 } }$$

Solve for x

$$x = \frac{5680}{19} = 298\frac{18}{19} \approx 298.947368421$$

$$x = \frac{5680}{21} = 270\frac{10}{21} \approx 270.476190476$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 284 since division by zero is not defined. Multiply both sides of the equation by \left(x-284\right)^{2}.

$$400\left(x-284\right)^{2}=x^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-284\right)^{2}.

$$400\left(x^{2}-568x+80656\right)=x^{2}$$

Use the distributive property to multiply 400 by x^{2}-568x+80656.

$$400x^{2}-227200x+32262400=x^{2}$$

Subtract x^{2} from both sides.

$$400x^{2}-227200x+32262400-x^{2}=0$$

Combine 400x^{2} and -x^{2} to get 399x^{2}.

$$399x^{2}-227200x+32262400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 399 for a, -227200 for b, and 32262400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-227200\right)±\sqrt{\left(-227200\right)^{2}-4\times 399\times 32262400}}{2\times 399}$$

Square -227200.

$$x=\frac{-\left(-227200\right)±\sqrt{51619840000-4\times 399\times 32262400}}{2\times 399}$$

Multiply -4 times 399.

$$x=\frac{-\left(-227200\right)±\sqrt{51619840000-1596\times 32262400}}{2\times 399}$$

Multiply -1596 times 32262400.

$$x=\frac{-\left(-227200\right)±\sqrt{51619840000-51490790400}}{2\times 399}$$

Add 51619840000 to -51490790400.

$$x=\frac{-\left(-227200\right)±\sqrt{129049600}}{2\times 399}$$

Take the square root of 129049600.

$$x=\frac{-\left(-227200\right)±11360}{2\times 399}$$

The opposite of -227200 is 227200.

$$x=\frac{227200±11360}{2\times 399}$$

Multiply 2 times 399.

$$x=\frac{227200±11360}{798}$$

Now solve the equation x=\frac{227200±11360}{798} when ± is plus. Add 227200 to 11360.

$$x=\frac{238560}{798}$$

Reduce the fraction \frac{238560}{798} to lowest terms by extracting and canceling out 42.

$$x=\frac{5680}{19}$$

Now solve the equation x=\frac{227200±11360}{798} when ± is minus. Subtract 11360 from 227200.

$$x=\frac{215840}{798}$$

Reduce the fraction \frac{215840}{798} to lowest terms by extracting and canceling out 38.

$$x=\frac{5680}{21}$$

The equation is now solved.

$$x=\frac{5680}{19} x=\frac{5680}{21}$$

Steps for Completing the Square

Variable x cannot be equal to 284 since division by zero is not defined. Multiply both sides of the equation by \left(x-284\right)^{2}.

$$400\left(x-284\right)^{2}=x^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-284\right)^{2}.

$$400\left(x^{2}-568x+80656\right)=x^{2}$$

Use the distributive property to multiply 400 by x^{2}-568x+80656.

$$400x^{2}-227200x+32262400=x^{2}$$

Subtract x^{2} from both sides.

$$400x^{2}-227200x+32262400-x^{2}=0$$

Combine 400x^{2} and -x^{2} to get 399x^{2}.

$$399x^{2}-227200x+32262400=0$$

Subtract 32262400 from both sides. Anything subtracted from zero gives its negation.

$$399x^{2}-227200x=-32262400$$

Divide both sides by 399.

$$\frac{399x^{2}-227200x}{399}=-\frac{32262400}{399}$$

Dividing by 399 undoes the multiplication by 399.

$$x^{2}-\frac{227200}{399}x=-\frac{32262400}{399}$$

Divide -\frac{227200}{399}, the coefficient of the x term, by 2 to get -\frac{113600}{399}. Then add the square of -\frac{113600}{399} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{227200}{399}x+\left(-\frac{113600}{399}\right)^{2}=-\frac{32262400}{399}+\left(-\frac{113600}{399}\right)^{2}$$

Square -\frac{113600}{399} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{227200}{399}x+\frac{12904960000}{159201}=-\frac{32262400}{399}+\frac{12904960000}{159201}$$

Add -\frac{32262400}{399} to \frac{12904960000}{159201} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{227200}{399}x+\frac{12904960000}{159201}=\frac{32262400}{159201}$$

Factor x^{2}-\frac{227200}{399}x+\frac{12904960000}{159201}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{113600}{399}\right)^{2}=\frac{32262400}{159201}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{113600}{399}\right)^{2}}=\sqrt{\frac{32262400}{159201}}$$

Simplify.

$$x-\frac{113600}{399}=\frac{5680}{399} x-\frac{113600}{399}=-\frac{5680}{399}$$

Add \frac{113600}{399} to both sides of the equation.

$$x=\frac{5680}{19} x=\frac{5680}{21}$$