Solve (102600+10000(0.416-x))(2-x)=102600*2

$$(102600+10000(0.416-x))(2-x)=102600 \times 2$$

Solve for x

$$x=\frac{3169-3\sqrt{1092729}}{500}\approx 0.065979273$$

$$x = \frac{3 \sqrt{1092729} + 3169}{500} \approx 12.610020727$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10000 by 0.416-x.

$$\left(102600+4160-10000x\right)\left(2-x\right)=102600\times 2$$

Add 102600 and 4160 to get 106760.

$$\left(106760-10000x\right)\left(2-x\right)=102600\times 2$$

Apply the distributive property by multiplying each term of 106760-10000x by each term of 2-x.

$$213520-106760x-20000x+10000x^{2}=102600\times 2$$

Combine -106760x and -20000x to get -126760x.

$$213520-126760x+10000x^{2}=102600\times 2$$

Multiply 102600 and 2 to get 205200.

$$213520-126760x+10000x^{2}=205200$$

Subtract 205200 from both sides.

$$213520-126760x+10000x^{2}-205200=0$$

Subtract 205200 from 213520 to get 8320.

$$8320-126760x+10000x^{2}=0$$

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.

$$10000x^{2}-126760x+8320=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10000 for a, -126760 for b, and 8320 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-126760\right)±\sqrt{\left(-126760\right)^{2}-4\times 10000\times 8320}}{2\times 10000}$$

Square -126760.

$$x=\frac{-\left(-126760\right)±\sqrt{16068097600-4\times 10000\times 8320}}{2\times 10000}$$

Multiply -4 times 10000.

$$x=\frac{-\left(-126760\right)±\sqrt{16068097600-40000\times 8320}}{2\times 10000}$$

Multiply -40000 times 8320.

$$x=\frac{-\left(-126760\right)±\sqrt{16068097600-332800000}}{2\times 10000}$$

Add 16068097600 to -332800000.

$$x=\frac{-\left(-126760\right)±\sqrt{15735297600}}{2\times 10000}$$

Take the square root of 15735297600.

$$x=\frac{-\left(-126760\right)±120\sqrt{1092729}}{2\times 10000}$$

The opposite of -126760 is 126760.

$$x=\frac{126760±120\sqrt{1092729}}{2\times 10000}$$

Multiply 2 times 10000.

$$x=\frac{126760±120\sqrt{1092729}}{20000}$$

Now solve the equation x=\frac{126760±120\sqrt{1092729}}{20000} when ± is plus. Add 126760 to 120\sqrt{1092729}.

$$x=\frac{120\sqrt{1092729}+126760}{20000}$$

Divide 126760+120\sqrt{1092729} by 20000.

$$x=\frac{3\sqrt{1092729}+3169}{500}$$

Now solve the equation x=\frac{126760±120\sqrt{1092729}}{20000} when ± is minus. Subtract 120\sqrt{1092729} from 126760.

$$x=\frac{126760-120\sqrt{1092729}}{20000}$$

Divide 126760-120\sqrt{1092729} by 20000.

$$x=\frac{3169-3\sqrt{1092729}}{500}$$

The equation is now solved.

$$x=\frac{3\sqrt{1092729}+3169}{500} x=\frac{3169-3\sqrt{1092729}}{500}$$

Steps for Completing the Square

Use the distributive property to multiply 10000 by 0.416-x.

$$\left(102600+4160-10000x\right)\left(2-x\right)=102600\times 2$$

Add 102600 and 4160 to get 106760.

$$\left(106760-10000x\right)\left(2-x\right)=102600\times 2$$

Apply the distributive property by multiplying each term of 106760-10000x by each term of 2-x.

$$213520-106760x-20000x+10000x^{2}=102600\times 2$$

Combine -106760x and -20000x to get -126760x.

$$213520-126760x+10000x^{2}=102600\times 2$$

Multiply 102600 and 2 to get 205200.

$$213520-126760x+10000x^{2}=205200$$

Subtract 213520 from both sides.

$$-126760x+10000x^{2}=205200-213520$$

Subtract 213520 from 205200 to get -8320.

$$-126760x+10000x^{2}=-8320$$

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.

$$10000x^{2}-126760x=-8320$$

Divide both sides by 10000.

$$\frac{10000x^{2}-126760x}{10000}=-\frac{8320}{10000}$$

Dividing by 10000 undoes the multiplication by 10000.

$$x^{2}+\left(-\frac{126760}{10000}\right)x=-\frac{8320}{10000}$$

Reduce the fraction \frac{-126760}{10000} to lowest terms by extracting and canceling out 40.

$$x^{2}-\frac{3169}{250}x=-\frac{8320}{10000}$$

Reduce the fraction \frac{-8320}{10000} to lowest terms by extracting and canceling out 80.

$$x^{2}-\frac{3169}{250}x=-\frac{104}{125}$$

Divide -\frac{3169}{250}, the coefficient of the x term, by 2 to get -\frac{3169}{500}. Then add the square of -\frac{3169}{500} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3169}{250}x+\left(-\frac{3169}{500}\right)^{2}=-\frac{104}{125}+\left(-\frac{3169}{500}\right)^{2}$$

Square -\frac{3169}{500} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3169}{250}x+\frac{10042561}{250000}=-\frac{104}{125}+\frac{10042561}{250000}$$

Add -\frac{104}{125} to \frac{10042561}{250000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3169}{250}x+\frac{10042561}{250000}=\frac{9834561}{250000}$$

Factor x^{2}-\frac{3169}{250}x+\frac{10042561}{250000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3169}{500}\right)^{2}=\frac{9834561}{250000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3169}{500}\right)^{2}}=\sqrt{\frac{9834561}{250000}}$$

Simplify.

$$x-\frac{3169}{500}=\frac{3\sqrt{1092729}}{500} x-\frac{3169}{500}=-\frac{3\sqrt{1092729}}{500}$$

Add \frac{3169}{500} to both sides of the equation.

$$x=\frac{3\sqrt{1092729}+3169}{500} x=\frac{3169-3\sqrt{1092729}}{500}$$