Solve 1302.13=1.22*(1300+x)*x

$$1302.13=1.22 \times (1300+x) \times x$$

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 1.22 by 1300+x.

$$1302.13=\left(1586+1.22x\right)x$$

Use the distributive property to multiply 1586+1.22x by x.

$$1302.13=1586x+1.22x^{2}$$

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

$$1586x+1.22x^{2}=1302.13$$

Subtract 1302.13 from both sides.

$$1586x+1.22x^{2}-1302.13=0$$

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.

$$1.22x^{2}+1586x-1302.13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.22 for a, 1586 for b, and -1302.13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1586±\sqrt{1586^{2}-4\times 1.22\left(-1302.13\right)}}{2\times 1.22}$$

Square 1586.

$$x=\frac{-1586±\sqrt{2515396-4\times 1.22\left(-1302.13\right)}}{2\times 1.22}$$

Multiply -4 times 1.22.

$$x=\frac{-1586±\sqrt{2515396-4.88\left(-1302.13\right)}}{2\times 1.22}$$

Multiply -4.88 times -1302.13 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1586±\sqrt{2515396+6354.3944}}{2\times 1.22}$$

Add 2515396 to 6354.3944.

$$x=\frac{-1586±\sqrt{2521750.3944}}{2\times 1.22}$$

Take the square root of 2521750.3944.

$$x=\frac{-1586±\frac{\sqrt{6304375986}}{50}}{2\times 1.22}$$

Multiply 2 times 1.22.

$$x=\frac{-1586±\frac{\sqrt{6304375986}}{50}}{2.44}$$

Now solve the equation x=\frac{-1586±\frac{\sqrt{6304375986}}{50}}{2.44} when ± is plus. Add -1586 to \frac{\sqrt{6304375986}}{50}.

$$x=\frac{\frac{\sqrt{6304375986}}{50}-1586}{2.44}$$

Divide -1586+\frac{\sqrt{6304375986}}{50} by 2.44 by multiplying -1586+\frac{\sqrt{6304375986}}{50} by the reciprocal of 2.44.

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

Now solve the equation x=\frac{-1586±\frac{\sqrt{6304375986}}{50}}{2.44} when ± is minus. Subtract \frac{\sqrt{6304375986}}{50} from -1586.

$$x=\frac{-\frac{\sqrt{6304375986}}{50}-1586}{2.44}$$

Divide -1586-\frac{\sqrt{6304375986}}{50} by 2.44 by multiplying -1586-\frac{\sqrt{6304375986}}{50} by the reciprocal of 2.44.

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

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 1.22 by 1300+x.

$$1302.13=\left(1586+1.22x\right)x$$

Use the distributive property to multiply 1586+1.22x by x.

$$1302.13=1586x+1.22x^{2}$$

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

$$1586x+1.22x^{2}=1302.13$$

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.

$$1.22x^{2}+1586x=1302.13$$

Divide both sides of the equation by 1.22, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.22x^{2}+1586x}{1.22}=\frac{1302.13}{1.22}$$

Dividing by 1.22 undoes the multiplication by 1.22.

$$x^{2}+\frac{1586}{1.22}x=\frac{1302.13}{1.22}$$

Divide 1586 by 1.22 by multiplying 1586 by the reciprocal of 1.22.

$$x^{2}+1300x=\frac{1302.13}{1.22}$$

Divide 1302.13 by 1.22 by multiplying 1302.13 by the reciprocal of 1.22.

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

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.

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