Solve 130213=122*(1300+x)*x

$$130213=122 \times (1300+x) \times x$$

Solve for x

$$x=\frac{\sqrt{6304375986}}{122}-650\approx 0.820497274$$

$$x=-\frac{\sqrt{6304375986}}{122}-650\approx -1300.820497274$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 122 by 1300+x.

$$130213=\left(158600+122x\right)x$$

Use the distributive property to multiply 158600+122x by x.

$$130213=158600x+122x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$158600x+122x^{2}=130213$$

Subtract 130213 from both sides.

$$158600x+122x^{2}-130213=0$$

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.

$$122x^{2}+158600x-130213=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 122 for a, 158600 for b, and -130213 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-158600±\sqrt{158600^{2}-4\times 122\left(-130213\right)}}{2\times 122}$$

Square 158600.

$$x=\frac{-158600±\sqrt{25153960000-4\times 122\left(-130213\right)}}{2\times 122}$$

Multiply -4 times 122.

$$x=\frac{-158600±\sqrt{25153960000-488\left(-130213\right)}}{2\times 122}$$

Multiply -488 times -130213.

$$x=\frac{-158600±\sqrt{25153960000+63543944}}{2\times 122}$$

Add 25153960000 to 63543944.

$$x=\frac{-158600±\sqrt{25217503944}}{2\times 122}$$

Take the square root of 25217503944.

$$x=\frac{-158600±2\sqrt{6304375986}}{2\times 122}$$

Multiply 2 times 122.

$$x=\frac{-158600±2\sqrt{6304375986}}{244}$$

Now solve the equation x=\frac{-158600±2\sqrt{6304375986}}{244} when ± is plus. Add -158600 to 2\sqrt{6304375986}.

$$x=\frac{2\sqrt{6304375986}-158600}{244}$$

Divide -158600+2\sqrt{6304375986} by 244.

$$x=\frac{\sqrt{6304375986}}{122}-650$$

Now solve the equation x=\frac{-158600±2\sqrt{6304375986}}{244} when ± is minus. Subtract 2\sqrt{6304375986} from -158600.

$$x=\frac{-2\sqrt{6304375986}-158600}{244}$$

Divide -158600-2\sqrt{6304375986} by 244.

$$x=-\frac{\sqrt{6304375986}}{122}-650$$

The equation is now solved.

$$x=\frac{\sqrt{6304375986}}{122}-650 x=-\frac{\sqrt{6304375986}}{122}-650$$

Steps for Completing the Square

Use the distributive property to multiply 122 by 1300+x.

$$130213=\left(158600+122x\right)x$$

Use the distributive property to multiply 158600+122x by x.

$$130213=158600x+122x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$158600x+122x^{2}=130213$$

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.

$$122x^{2}+158600x=130213$$

Divide both sides by 122.

$$\frac{122x^{2}+158600x}{122}=\frac{130213}{122}$$

Dividing by 122 undoes the multiplication by 122.

$$x^{2}+\frac{158600}{122}x=\frac{130213}{122}$$

Divide 158600 by 122.

$$x^{2}+1300x=\frac{130213}{122}$$

Divide 1300, the coefficient of the x term, by 2 to get 650. Then add the square of 650 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+1300x+650^{2}=\frac{130213}{122}+650^{2}$$

Square 650.

$$x^{2}+1300x+422500=\frac{130213}{122}+422500$$

Add \frac{130213}{122} to 422500.

$$x^{2}+1300x+422500=\frac{51675213}{122}$$

Factor x^{2}+1300x+422500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+650\right)^{2}=\frac{51675213}{122}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+650\right)^{2}}=\sqrt{\frac{51675213}{122}}$$

Simplify.

$$x+650=\frac{\sqrt{6304375986}}{122} x+650=-\frac{\sqrt{6304375986}}{122}$$

Subtract 650 from both sides of the equation.

$$x=\frac{\sqrt{6304375986}}{122}-650 x=-\frac{\sqrt{6304375986}}{122}-650$$