Solve 18000+130x+(5%x^2)=0

$$18000+130x+(5 \% { x }^{ 2 } )=0$$

Solve for x

$$x=100\sqrt{133}-1300\approx -146.743740533$$

$$x=-100\sqrt{133}-1300\approx -2453.256259467$$

Steps Using the Quadratic Formula

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.

$$\frac{1}{20}x^{2}+130x+18000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{20} for a, 130 for b, and 18000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-130±\sqrt{130^{2}-4\times \frac{1}{20}\times 18000}}{2\times \frac{1}{20}}$$

Square 130.

$$x=\frac{-130±\sqrt{16900-4\times \frac{1}{20}\times 18000}}{2\times \frac{1}{20}}$$

Multiply -4 times \frac{1}{20}.

$$x=\frac{-130±\sqrt{16900-\frac{1}{5}\times 18000}}{2\times \frac{1}{20}}$$

Multiply -\frac{1}{5} times 18000.

$$x=\frac{-130±\sqrt{16900-3600}}{2\times \frac{1}{20}}$$

Add 16900 to -3600.

$$x=\frac{-130±\sqrt{13300}}{2\times \frac{1}{20}}$$

Take the square root of 13300.

$$x=\frac{-130±10\sqrt{133}}{2\times \frac{1}{20}}$$

Multiply 2 times \frac{1}{20}.

$$x=\frac{-130±10\sqrt{133}}{\frac{1}{10}}$$

Now solve the equation x=\frac{-130±10\sqrt{133}}{\frac{1}{10}} when ± is plus. Add -130 to 10\sqrt{133}.

$$x=\frac{10\sqrt{133}-130}{\frac{1}{10}}$$

Divide -130+10\sqrt{133} by \frac{1}{10} by multiplying -130+10\sqrt{133} by the reciprocal of \frac{1}{10}.

$$x=100\sqrt{133}-1300$$

Now solve the equation x=\frac{-130±10\sqrt{133}}{\frac{1}{10}} when ± is minus. Subtract 10\sqrt{133} from -130.

$$x=\frac{-10\sqrt{133}-130}{\frac{1}{10}}$$

Divide -130-10\sqrt{133} by \frac{1}{10} by multiplying -130-10\sqrt{133} by the reciprocal of \frac{1}{10}.

$$x=-100\sqrt{133}-1300$$

The equation is now solved.

$$x=100\sqrt{133}-1300 x=-100\sqrt{133}-1300$$

Steps for Completing the Square

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.

$$\frac{1}{20}x^{2}+130x+18000=0$$

Subtract 18000 from both sides of the equation.

$$\frac{1}{20}x^{2}+130x+18000-18000=-18000$$

Subtracting 18000 from itself leaves 0.

$$\frac{1}{20}x^{2}+130x=-18000$$

Multiply both sides by 20.

$$\frac{\frac{1}{20}x^{2}+130x}{\frac{1}{20}}=-\frac{18000}{\frac{1}{20}}$$

Dividing by \frac{1}{20} undoes the multiplication by \frac{1}{20}.

$$x^{2}+\frac{130}{\frac{1}{20}}x=-\frac{18000}{\frac{1}{20}}$$

Divide 130 by \frac{1}{20} by multiplying 130 by the reciprocal of \frac{1}{20}.

$$x^{2}+2600x=-\frac{18000}{\frac{1}{20}}$$

Divide -18000 by \frac{1}{20} by multiplying -18000 by the reciprocal of \frac{1}{20}.

$$x^{2}+2600x=-360000$$

Divide 2600, the coefficient of the x term, by 2 to get 1300. Then add the square of 1300 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2600x+1300^{2}=-360000+1300^{2}$$

Square 1300.

$$x^{2}+2600x+1690000=-360000+1690000$$

Add -360000 to 1690000.

$$x^{2}+2600x+1690000=1330000$$

Factor x^{2}+2600x+1690000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1300\right)^{2}=1330000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1300\right)^{2}}=\sqrt{1330000}$$

Simplify.

$$x+1300=100\sqrt{133} x+1300=-100\sqrt{133}$$

Subtract 1300 from both sides of the equation.

$$x=100\sqrt{133}-1300 x=-100\sqrt{133}-1300$$