Solve 16x-100x+1/5x^2-3000=0

$$16x-100x+ \frac{ 1 }{ 5 } { x }^{ 2 } -3000=0$$

Solve for x

$$x=10\sqrt{591}+210\approx 453.104915623$$

$$x=210-10\sqrt{591}\approx -33.104915623$$

Steps Using the Quadratic Formula

Combine 16x and -100x to get -84x.

$$-84x+\frac{1}{5}x^{2}-3000=0$$

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.

$$\frac{1}{5}x^{2}-84x-3000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{5} for a, -84 for b, and -3000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-84\right)±\sqrt{\left(-84\right)^{2}-4\times \frac{1}{5}\left(-3000\right)}}{2\times \frac{1}{5}}$$

Square -84.

$$x=\frac{-\left(-84\right)±\sqrt{7056-4\times \frac{1}{5}\left(-3000\right)}}{2\times \frac{1}{5}}$$

Multiply -4 times \frac{1}{5}.

$$x=\frac{-\left(-84\right)±\sqrt{7056-\frac{4}{5}\left(-3000\right)}}{2\times \frac{1}{5}}$$

Multiply -\frac{4}{5} times -3000.

$$x=\frac{-\left(-84\right)±\sqrt{7056+2400}}{2\times \frac{1}{5}}$$

Add 7056 to 2400.

$$x=\frac{-\left(-84\right)±\sqrt{9456}}{2\times \frac{1}{5}}$$

Take the square root of 9456.

$$x=\frac{-\left(-84\right)±4\sqrt{591}}{2\times \frac{1}{5}}$$

The opposite of -84 is 84.

$$x=\frac{84±4\sqrt{591}}{2\times \frac{1}{5}}$$

Multiply 2 times \frac{1}{5}.

$$x=\frac{84±4\sqrt{591}}{\frac{2}{5}}$$

Now solve the equation x=\frac{84±4\sqrt{591}}{\frac{2}{5}} when ± is plus. Add 84 to 4\sqrt{591}.

$$x=\frac{4\sqrt{591}+84}{\frac{2}{5}}$$

Divide 84+4\sqrt{591} by \frac{2}{5} by multiplying 84+4\sqrt{591} by the reciprocal of \frac{2}{5}.

$$x=10\sqrt{591}+210$$

Now solve the equation x=\frac{84±4\sqrt{591}}{\frac{2}{5}} when ± is minus. Subtract 4\sqrt{591} from 84.

$$x=\frac{84-4\sqrt{591}}{\frac{2}{5}}$$

Divide 84-4\sqrt{591} by \frac{2}{5} by multiplying 84-4\sqrt{591} by the reciprocal of \frac{2}{5}.

$$x=210-10\sqrt{591}$$

The equation is now solved.

$$x=10\sqrt{591}+210 x=210-10\sqrt{591}$$

Steps for Completing the Square

Combine 16x and -100x to get -84x.

$$-84x+\frac{1}{5}x^{2}-3000=0$$

Add 3000 to both sides. Anything plus zero gives itself.

$$-84x+\frac{1}{5}x^{2}=3000$$

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.

$$\frac{1}{5}x^{2}-84x=3000$$

Multiply both sides by 5.

$$\frac{\frac{1}{5}x^{2}-84x}{\frac{1}{5}}=\frac{3000}{\frac{1}{5}}$$

Dividing by \frac{1}{5} undoes the multiplication by \frac{1}{5}.

$$x^{2}+\left(-\frac{84}{\frac{1}{5}}\right)x=\frac{3000}{\frac{1}{5}}$$

Divide -84 by \frac{1}{5} by multiplying -84 by the reciprocal of \frac{1}{5}.

$$x^{2}-420x=\frac{3000}{\frac{1}{5}}$$

Divide 3000 by \frac{1}{5} by multiplying 3000 by the reciprocal of \frac{1}{5}.

$$x^{2}-420x=15000$$

Divide -420, the coefficient of the x term, by 2 to get -210. Then add the square of -210 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-420x+\left(-210\right)^{2}=15000+\left(-210\right)^{2}$$

Square -210.

$$x^{2}-420x+44100=15000+44100$$

Add 15000 to 44100.

$$x^{2}-420x+44100=59100$$

Factor x^{2}-420x+44100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-210\right)^{2}=59100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-210\right)^{2}}=\sqrt{59100}$$

Simplify.

$$x-210=10\sqrt{591} x-210=-10\sqrt{591}$$

Add 210 to both sides of the equation.

$$x=10\sqrt{591}+210 x=210-10\sqrt{591}$$