Solve 30*10^6=14.3*400*(125-x/2)*x

$$30 \times 10 ^ { 6 } = 14.3 \times 400 \times ( 125 - \frac { x } { 2 } ) \cdot x$$

Solve for x

$$x=\frac{125\sqrt{6721}}{143}+125\approx 196.662329787$$

$$x=-\frac{125\sqrt{6721}}{143}+125\approx 53.337670213$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$60\times 10^{6}=28.6\times 400\left(125-\frac{x}{2}\right)x$$

Calculate 10 to the power of 6 and get 1000000.

$$60\times 1000000=28.6\times 400\left(125-\frac{x}{2}\right)x$$

Multiply 60 and 1000000 to get 60000000.

$$60000000=28.6\times 400\left(125-\frac{x}{2}\right)x$$

Multiply 28.6 and 400 to get 11440.

$$60000000=11440\left(125-\frac{x}{2}\right)x$$

Use the distributive property to multiply 11440 by 125-\frac{x}{2}.

$$60000000=\left(1430000+11440\left(-\frac{x}{2}\right)\right)x$$

Cancel out 2, the greatest common factor in 11440 and 2.

$$60000000=\left(1430000-5720x\right)x$$

Use the distributive property to multiply 1430000-5720x by x.

$$60000000=1430000x-5720x^{2}$$

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

$$1430000x-5720x^{2}=60000000$$

Subtract 60000000 from both sides.

$$1430000x-5720x^{2}-60000000=0$$

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.

$$-5720x^{2}+1430000x-60000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5720 for a, 1430000 for b, and -60000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1430000±\sqrt{1430000^{2}-4\left(-5720\right)\left(-60000000\right)}}{2\left(-5720\right)}$$

Square 1430000.

$$x=\frac{-1430000±\sqrt{2044900000000-4\left(-5720\right)\left(-60000000\right)}}{2\left(-5720\right)}$$

Multiply -4 times -5720.

$$x=\frac{-1430000±\sqrt{2044900000000+22880\left(-60000000\right)}}{2\left(-5720\right)}$$

Multiply 22880 times -60000000.

$$x=\frac{-1430000±\sqrt{2044900000000-1372800000000}}{2\left(-5720\right)}$$

Add 2044900000000 to -1372800000000.

$$x=\frac{-1430000±\sqrt{672100000000}}{2\left(-5720\right)}$$

Take the square root of 672100000000.

$$x=\frac{-1430000±10000\sqrt{6721}}{2\left(-5720\right)}$$

Multiply 2 times -5720.

$$x=\frac{-1430000±10000\sqrt{6721}}{-11440}$$

Now solve the equation x=\frac{-1430000±10000\sqrt{6721}}{-11440} when ± is plus. Add -1430000 to 10000\sqrt{6721}.

$$x=\frac{10000\sqrt{6721}-1430000}{-11440}$$

Divide -1430000+10000\sqrt{6721} by -11440.

$$x=-\frac{125\sqrt{6721}}{143}+125$$

Now solve the equation x=\frac{-1430000±10000\sqrt{6721}}{-11440} when ± is minus. Subtract 10000\sqrt{6721} from -1430000.

$$x=\frac{-10000\sqrt{6721}-1430000}{-11440}$$

Divide -1430000-10000\sqrt{6721} by -11440.

$$x=\frac{125\sqrt{6721}}{143}+125$$

The equation is now solved.

$$x=-\frac{125\sqrt{6721}}{143}+125 x=\frac{125\sqrt{6721}}{143}+125$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$60\times 10^{6}=28.6\times 400\left(125-\frac{x}{2}\right)x$$

Calculate 10 to the power of 6 and get 1000000.

$$60\times 1000000=28.6\times 400\left(125-\frac{x}{2}\right)x$$

Multiply 60 and 1000000 to get 60000000.

$$60000000=28.6\times 400\left(125-\frac{x}{2}\right)x$$

Multiply 28.6 and 400 to get 11440.

$$60000000=11440\left(125-\frac{x}{2}\right)x$$

Use the distributive property to multiply 11440 by 125-\frac{x}{2}.

$$60000000=\left(1430000+11440\left(-\frac{x}{2}\right)\right)x$$

Cancel out 2, the greatest common factor in 11440 and 2.

$$60000000=\left(1430000-5720x\right)x$$

Use the distributive property to multiply 1430000-5720x by x.

$$60000000=1430000x-5720x^{2}$$

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

$$1430000x-5720x^{2}=60000000$$

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.

$$-5720x^{2}+1430000x=60000000$$

Divide both sides by -5720.

$$\frac{-5720x^{2}+1430000x}{-5720}=\frac{60000000}{-5720}$$

Dividing by -5720 undoes the multiplication by -5720.

$$x^{2}+\frac{1430000}{-5720}x=\frac{60000000}{-5720}$$

Divide 1430000 by -5720.

$$x^{2}-250x=\frac{60000000}{-5720}$$

Reduce the fraction \frac{60000000}{-5720} to lowest terms by extracting and canceling out 40.

$$x^{2}-250x=-\frac{1500000}{143}$$

Divide -250, the coefficient of the x term, by 2 to get -125. Then add the square of -125 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-250x+\left(-125\right)^{2}=-\frac{1500000}{143}+\left(-125\right)^{2}$$

Square -125.

$$x^{2}-250x+15625=-\frac{1500000}{143}+15625$$

Add -\frac{1500000}{143} to 15625.

$$x^{2}-250x+15625=\frac{734375}{143}$$

Factor x^{2}-250x+15625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-125\right)^{2}=\frac{734375}{143}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-125\right)^{2}}=\sqrt{\frac{734375}{143}}$$

Simplify.

$$x-125=\frac{125\sqrt{6721}}{143} x-125=-\frac{125\sqrt{6721}}{143}$$

Add 125 to both sides of the equation.

$$x=\frac{125\sqrt{6721}}{143}+125 x=-\frac{125\sqrt{6721}}{143}+125$$