Solve 1300*x^2/2=9*5600*(406-x)

$$1300 \times x ^ { 2 } / 2 = 9 \times 5600 \times ( 406 - x )$$

Solve for x

$$x = \frac{84 \sqrt{790} - 504}{13} \approx 142.844834322$$

$$x=\frac{-84\sqrt{790}-504}{13}\approx -220.383295861$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$1300x^{2}=18\times 5600\left(406-x\right)$$

Multiply 18 and 5600 to get 100800.

$$1300x^{2}=100800\left(406-x\right)$$

Use the distributive property to multiply 100800 by 406-x.

$$1300x^{2}=40924800-100800x$$

Subtract 40924800 from both sides.

$$1300x^{2}-40924800=-100800x$$

Add 100800x to both sides.

$$1300x^{2}-40924800+100800x=0$$

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.

$$1300x^{2}+100800x-40924800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1300 for a, 100800 for b, and -40924800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100800±\sqrt{100800^{2}-4\times 1300\left(-40924800\right)}}{2\times 1300}$$

Square 100800.

$$x=\frac{-100800±\sqrt{10160640000-4\times 1300\left(-40924800\right)}}{2\times 1300}$$

Multiply -4 times 1300.

$$x=\frac{-100800±\sqrt{10160640000-5200\left(-40924800\right)}}{2\times 1300}$$

Multiply -5200 times -40924800.

$$x=\frac{-100800±\sqrt{10160640000+212808960000}}{2\times 1300}$$

Add 10160640000 to 212808960000.

$$x=\frac{-100800±\sqrt{222969600000}}{2\times 1300}$$

Take the square root of 222969600000.

$$x=\frac{-100800±16800\sqrt{790}}{2\times 1300}$$

Multiply 2 times 1300.

$$x=\frac{-100800±16800\sqrt{790}}{2600}$$

Now solve the equation x=\frac{-100800±16800\sqrt{790}}{2600} when ± is plus. Add -100800 to 16800\sqrt{790}.

$$x=\frac{16800\sqrt{790}-100800}{2600}$$

Divide -100800+16800\sqrt{790} by 2600.

$$x=\frac{84\sqrt{790}-504}{13}$$

Now solve the equation x=\frac{-100800±16800\sqrt{790}}{2600} when ± is minus. Subtract 16800\sqrt{790} from -100800.

$$x=\frac{-16800\sqrt{790}-100800}{2600}$$

Divide -100800-16800\sqrt{790} by 2600.

$$x=\frac{-84\sqrt{790}-504}{13}$$

The equation is now solved.

$$x=\frac{84\sqrt{790}-504}{13} x=\frac{-84\sqrt{790}-504}{13}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$1300x^{2}=18\times 5600\left(406-x\right)$$

Multiply 18 and 5600 to get 100800.

$$1300x^{2}=100800\left(406-x\right)$$

Use the distributive property to multiply 100800 by 406-x.

$$1300x^{2}=40924800-100800x$$

Add 100800x to both sides.

$$1300x^{2}+100800x=40924800$$

Divide both sides by 1300.

$$\frac{1300x^{2}+100800x}{1300}=\frac{40924800}{1300}$$

Dividing by 1300 undoes the multiplication by 1300.

$$x^{2}+\frac{100800}{1300}x=\frac{40924800}{1300}$$

Reduce the fraction \frac{100800}{1300} to lowest terms by extracting and canceling out 100.

$$x^{2}+\frac{1008}{13}x=\frac{40924800}{1300}$$

Reduce the fraction \frac{40924800}{1300} to lowest terms by extracting and canceling out 100.

$$x^{2}+\frac{1008}{13}x=\frac{409248}{13}$$

Divide \frac{1008}{13}, the coefficient of the x term, by 2 to get \frac{504}{13}. Then add the square of \frac{504}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1008}{13}x+\left(\frac{504}{13}\right)^{2}=\frac{409248}{13}+\left(\frac{504}{13}\right)^{2}$$

Square \frac{504}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1008}{13}x+\frac{254016}{169}=\frac{409248}{13}+\frac{254016}{169}$$

Add \frac{409248}{13} to \frac{254016}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1008}{13}x+\frac{254016}{169}=\frac{5574240}{169}$$

Factor x^{2}+\frac{1008}{13}x+\frac{254016}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{504}{13}\right)^{2}=\frac{5574240}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{504}{13}\right)^{2}}=\sqrt{\frac{5574240}{169}}$$

Simplify.

$$x+\frac{504}{13}=\frac{84\sqrt{790}}{13} x+\frac{504}{13}=-\frac{84\sqrt{790}}{13}$$

Subtract \frac{504}{13} from both sides of the equation.

$$x=\frac{84\sqrt{790}-504}{13} x=\frac{-84\sqrt{790}-504}{13}$$