Solve 14.3*250left(385-x/2right)x+360*402left(385-35right)=230000000

$$14.3 \cdot 250 \left( 385- \frac{ x }{ 2 } \right) x+360 \cdot 402 \left( 385-35 \right) = 230000000$$

Solve for x

$$x=\frac{\sqrt{5794745}}{11}+385\approx 603.838878048$$

$$x=-\frac{\sqrt{5794745}}{11}+385\approx 166.161121952$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$28.6\times 250\left(385-\frac{x}{2}\right)x+720\times 402\left(385-35\right)=460000000$$

Multiply 28.6 and 250 to get 7150.

$$7150\left(385-\frac{x}{2}\right)x+720\times 402\left(385-35\right)=460000000$$

Use the distributive property to multiply 7150 by 385-\frac{x}{2}.

$$\left(2752750+7150\left(-\frac{x}{2}\right)\right)x+720\times 402\left(385-35\right)=460000000$$

Cancel out 2, the greatest common factor in 7150 and 2.

$$\left(2752750-3575x\right)x+720\times 402\left(385-35\right)=460000000$$

Use the distributive property to multiply 2752750-3575x by x.

$$2752750x-3575x^{2}+720\times 402\left(385-35\right)=460000000$$

Multiply 720 and 402 to get 289440.

$$2752750x-3575x^{2}+289440\left(385-35\right)=460000000$$

Subtract 35 from 385 to get 350.

$$2752750x-3575x^{2}+289440\times 350=460000000$$

Multiply 289440 and 350 to get 101304000.

$$2752750x-3575x^{2}[phone]=460000000$$

Subtract 460000000 from both sides.

$$2752750x-3575x^{2}[phone]-460000000=0$$

Subtract 460000000 from 101304000 to get -358696000.

$$2752750x-3575x^{2}-358696000=0$$

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.

$$-3575x^{2}+2752750x-358696000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3575 for a, 2752750 for b, and -358696000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2752750±\sqrt{2752750^{2}-4\left(-3575\right)\left(-358696000\right)}}{2\left(-3575\right)}$$

Square 2752750.

$$x=\frac{-2752750±\sqrt{7577632562500-4\left(-3575\right)\left(-358696000\right)}}{2\left(-3575\right)}$$

Multiply -4 times -3575.

$$x=\frac{-2752750±\sqrt{7577632562500+14300\left(-358696000\right)}}{2\left(-3575\right)}$$

Multiply 14300 times -358696000.

$$x=\frac{-2752750±\sqrt{7577632562500-5129352800000}}{2\left(-3575\right)}$$

Add 7577632562500 to -5129352800000.

$$x=\frac{-2752750±\sqrt{2448279762500}}{2\left(-3575\right)}$$

Take the square root of 2448279762500.

$$x=\frac{-2752750±650\sqrt{5794745}}{2\left(-3575\right)}$$

Multiply 2 times -3575.

$$x=\frac{-2752750±650\sqrt{5794745}}{-7150}$$

Now solve the equation x=\frac{-2752750±650\sqrt{5794745}}{-7150} when ± is plus. Add -2752750 to 650\sqrt{5794745}.

$$x=\frac{650\sqrt{5794745}-2752750}{-7150}$$

Divide -2752750+650\sqrt{5794745} by -7150.

$$x=-\frac{\sqrt{5794745}}{11}+385$$

Now solve the equation x=\frac{-2752750±650\sqrt{5794745}}{-7150} when ± is minus. Subtract 650\sqrt{5794745} from -2752750.

$$x=\frac{-650\sqrt{5794745}-2752750}{-7150}$$

Divide -2752750-650\sqrt{5794745} by -7150.

$$x=\frac{\sqrt{5794745}}{11}+385$$

The equation is now solved.

$$x=-\frac{\sqrt{5794745}}{11}+385 x=\frac{\sqrt{5794745}}{11}+385$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$28.6\times 250\left(385-\frac{x}{2}\right)x+720\times 402\left(385-35\right)=460000000$$

Multiply 28.6 and 250 to get 7150.

$$7150\left(385-\frac{x}{2}\right)x+720\times 402\left(385-35\right)=460000000$$

Use the distributive property to multiply 7150 by 385-\frac{x}{2}.

$$\left(2752750+7150\left(-\frac{x}{2}\right)\right)x+720\times 402\left(385-35\right)=460000000$$

Cancel out 2, the greatest common factor in 7150 and 2.

$$\left(2752750-3575x\right)x+720\times 402\left(385-35\right)=460000000$$

Use the distributive property to multiply 2752750-3575x by x.

$$2752750x-3575x^{2}+720\times 402\left(385-35\right)=460000000$$

Multiply 720 and 402 to get 289440.

$$2752750x-3575x^{2}+289440\left(385-35\right)=460000000$$

Subtract 35 from 385 to get 350.

$$2752750x-3575x^{2}+289440\times 350=460000000$$

Multiply 289440 and 350 to get 101304000.

$$2752750x-3575x^{2}[phone]=460000000$$

Subtract 101304000 from both sides.

$$2752750x-3575x^{2}=460000000-101304000$$

Subtract 101304000 from 460000000 to get 358696000.

$$2752750x-3575x^{2}=358696000$$

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.

$$-3575x^{2}+2752750x=358696000$$

Divide both sides by -3575.

$$\frac{-3575x^{2}+2752750x}{-3575}=\frac{358696000}{-3575}$$

Dividing by -3575 undoes the multiplication by -3575.

$$x^{2}+\frac{2752750}{-3575}x=\frac{358696000}{-3575}$$

Divide 2752750 by -3575.

$$x^{2}-770x=\frac{358696000}{-3575}$$

Reduce the fraction \frac{358696000}{-3575} to lowest terms by extracting and canceling out 325.

$$x^{2}-770x=-\frac{1103680}{11}$$

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.

$$x^{2}-770x+\left(-385\right)^{2}=-\frac{1103680}{11}+\left(-385\right)^{2}$$

Square -385.

$$x^{2}-770x+148225=-\frac{1103680}{11}+148225$$

Add -\frac{1103680}{11} to 148225.

$$x^{2}-770x+148225=\frac{526795}{11}$$

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}.

$$\left(x-385\right)^{2}=\frac{526795}{11}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-385\right)^{2}}=\sqrt{\frac{526795}{11}}$$

Simplify.

$$x-385=\frac{\sqrt{5794745}}{11} x-385=-\frac{\sqrt{5794745}}{11}$$

Add 385 to both sides of the equation.

$$x=\frac{\sqrt{5794745}}{11}+385 x=-\frac{\sqrt{5794745}}{11}+385$$