Solve 3565(385-x/2)x=230000000

$$3565 ( 385 - \frac { x } { 2 } ) x = 230000000$$

Solve for x

$$x=\frac{5\sqrt{737769}}{31}+385\approx 523.537871846$$

$$x=-\frac{5\sqrt{737769}}{31}+385\approx 246.462128154$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$7130\left(385-\frac{x}{2}\right)x=460000000$$

Use the distributive property to multiply 7130 by 385-\frac{x}{2}.

$$\left(2745050+7130\left(-\frac{x}{2}\right)\right)x=460000000$$

Cancel out 2, the greatest common factor in 7130 and 2.

$$\left(2745050-3565x\right)x=460000000$$

Use the distributive property to multiply 2745050-3565x by x.

$$2745050x-3565x^{2}=460000000$$

Subtract 460000000 from both sides.

$$2745050x-3565x^{2}-460000000=0$$

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.

$$-3565x^{2}+2745050x-460000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3565 for a, 2745050 for b, and -460000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2745050±\sqrt{2745050^{2}-4\left(-3565\right)\left(-460000000\right)}}{2\left(-3565\right)}$$

Square 2745050.

$$x=\frac{-2745050±\sqrt{7535299502500-4\left(-3565\right)\left(-460000000\right)}}{2\left(-3565\right)}$$

Multiply -4 times -3565.

$$x=\frac{-2745050±\sqrt{7535299502500+14260\left(-460000000\right)}}{2\left(-3565\right)}$$

Multiply 14260 times -460000000.

$$x=\frac{-2745050±\sqrt{7535299502500-6559600000000}}{2\left(-3565\right)}$$

Add 7535299502500 to -6559600000000.

$$x=\frac{-2745050±\sqrt{975699502500}}{2\left(-3565\right)}$$

Take the square root of 975699502500.

$$x=\frac{-2745050±1150\sqrt{737769}}{2\left(-3565\right)}$$

Multiply 2 times -3565.

$$x=\frac{-2745050±1150\sqrt{737769}}{-7130}$$

Now solve the equation x=\frac{-2745050±1150\sqrt{737769}}{-7130} when ± is plus. Add -2745050 to 1150\sqrt{737769}.

$$x=\frac{1150\sqrt{737769}-2745050}{-7130}$$

Divide -2745050+1150\sqrt{737769} by -7130.

$$x=-\frac{5\sqrt{737769}}{31}+385$$

Now solve the equation x=\frac{-2745050±1150\sqrt{737769}}{-7130} when ± is minus. Subtract 1150\sqrt{737769} from -2745050.

$$x=\frac{-1150\sqrt{737769}-2745050}{-7130}$$

Divide -2745050-1150\sqrt{737769} by -7130.

$$x=\frac{5\sqrt{737769}}{31}+385$$

The equation is now solved.

$$x=-\frac{5\sqrt{737769}}{31}+385 x=\frac{5\sqrt{737769}}{31}+385$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$7130\left(385-\frac{x}{2}\right)x=460000000$$

Use the distributive property to multiply 7130 by 385-\frac{x}{2}.

$$\left(2745050+7130\left(-\frac{x}{2}\right)\right)x=460000000$$

Cancel out 2, the greatest common factor in 7130 and 2.

$$\left(2745050-3565x\right)x=460000000$$

Use the distributive property to multiply 2745050-3565x by x.

$$2745050x-3565x^{2}=460000000$$

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.

$$-3565x^{2}+2745050x=460000000$$

Divide both sides by -3565.

$$\frac{-3565x^{2}+2745050x}{-3565}=\frac{460000000}{-3565}$$

Dividing by -3565 undoes the multiplication by -3565.

$$x^{2}+\frac{2745050}{-3565}x=\frac{460000000}{-3565}$$

Divide 2745050 by -3565.

$$x^{2}-770x=\frac{460000000}{-3565}$$

Reduce the fraction \frac{460000000}{-3565} to lowest terms by extracting and canceling out 115.

$$x^{2}-770x=-\frac{4000000}{31}$$

Divide -770, the coefficient of the x term, by 2 to get -385. Then add the square of -385 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-770x+\left(-385\right)^{2}=-\frac{4000000}{31}+\left(-385\right)^{2}$$

Square -385.

$$x^{2}-770x+148225=-\frac{4000000}{31}+148225$$

Add -\frac{4000000}{31} to 148225.

$$x^{2}-770x+148225=\frac{594975}{31}$$

Factor x^{2}-770x+148225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-385\right)^{2}=\frac{594975}{31}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-385\right)^{2}}=\sqrt{\frac{594975}{31}}$$

Simplify.

$$x-385=\frac{5\sqrt{737769}}{31} x-385=-\frac{5\sqrt{737769}}{31}$$

Add 385 to both sides of the equation.

$$x=\frac{5\sqrt{737769}}{31}+385 x=-\frac{5\sqrt{737769}}{31}+385$$