Solve 4*x+100000/x+102500=11195*x/x+100000

$$4 \cdot \frac { x + 100000 } { x + 102500 } = 11195 \cdot \frac { x } { x + 100000 }$$

Solve for x

$$x = \frac{6250 \sqrt{8426769809} - 573343750}{11191} \approx 34.871219685$$

$$x=\frac{-6250\sqrt{8426769809}-573343750}{11191}\approx -102500.021787105$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -102500,-100000 since division by zero is not defined. Multiply both sides of the equation by \left(x+100000\right)\left(x+102500\right), the least common multiple of x+102500,x+100000.

$$4\left(x+100000\right)\left(x+100000\right)=11195\left(x+102500\right)x$$

Multiply x+100000 and x+100000 to get \left(x+100000\right)^{2}.

$$4\left(x+100000\right)^{2}=11195\left(x+102500\right)x$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+100000\right)^{2}.

$$4\left(x^{2}+200000x+10000000000\right)=11195\left(x+102500\right)x$$

Use the distributive property to multiply 4 by x^{2}+200000x+10000000000.

$$4x^{2}+800000x+40000000000=11195\left(x+102500\right)x$$

Use the distributive property to multiply 11195 by x+102500.

$$4x^{2}+800000x+40000000000=\left(11195x+1147487500\right)x$$

Use the distributive property to multiply 11195x+1147487500 by x.

$$4x^{2}+800000x+40000000000=11195x^{2}[phone]x$$

Subtract 11195x^{2} from both sides.

$$4x^{2}+800000x+40000000000-11195x^{2}=1147487500x$$

Combine 4x^{2} and -11195x^{2} to get -11191x^{2}.

$$-11191x^{2}+800000x+40000000000=1147487500x$$

Subtract 1147487500x from both sides.

$$-11191x^{2}+800000x+40000000000-1147487500x=0$$

Combine 800000x and -1147487500x to get -1146687500x.

$$-11191x^{2}-1146687500x+40000000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -11191 for a, -1146687500 for b, and 40000000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1146687500\right)±\sqrt{\left(-1146687500\right)^{2}-4\left(-11191\right)\times 40000000000}}{2\left(-11191\right)}$$

Square -1146687500.

$$x=\frac{-\left(-1146687500\right)±\sqrt{1314892222656250000-4\left(-11191\right)\times 40000000000}}{2\left(-11191\right)}$$

Multiply -4 times -11191.

$$x=\frac{-\left(-1146687500\right)±\sqrt{1314892222656250000+44764\times 40000000000}}{2\left(-11191\right)}$$

Multiply 44764 times 40000000000.

$$x=\frac{-\left(-1146687500\right)±\sqrt{1314892222656250000+1790560000000000}}{2\left(-11191\right)}$$

Add 1314892222656250000 to 1790560000000000.

$$x=\frac{-\left(-1146687500\right)±\sqrt{1316682782656250000}}{2\left(-11191\right)}$$

Take the square root of 1316682782656250000.

$$x=\frac{-\left(-1146687500\right)±12500\sqrt{8426769809}}{2\left(-11191\right)}$$

The opposite of -1146687500 is 1146687500.

$$x=\frac{1146687500±12500\sqrt{8426769809}}{2\left(-11191\right)}$$

Multiply 2 times -11191.

$$x=\frac{1146687500±12500\sqrt{8426769809}}{-22382}$$

Now solve the equation x=\frac{1146687500±12500\sqrt{8426769809}}{-22382} when ± is plus. Add 1146687500 to 12500\sqrt{8426769809}.

$$x=\frac{12500\sqrt{8426769809}[phone]}{-22382}$$

Divide 1146687500+12500\sqrt{8426769809} by -22382.

$$x=\frac{-6250\sqrt{8426769809}-573343750}{11191}$$

Now solve the equation x=\frac{1146687500±12500\sqrt{8426769809}}{-22382} when ± is minus. Subtract 12500\sqrt{8426769809} from 1146687500.

$$x=\frac{1146687500-12500\sqrt{8426769809}}{-22382}$$

Divide 1146687500-12500\sqrt{8426769809} by -22382.

$$x=\frac{6250\sqrt{8426769809}-573343750}{11191}$$

The equation is now solved.

$$x=\frac{-6250\sqrt{8426769809}-573343750}{11191} x=\frac{6250\sqrt{8426769809}-573343750}{11191}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -102500,-100000 since division by zero is not defined. Multiply both sides of the equation by \left(x+100000\right)\left(x+102500\right), the least common multiple of x+102500,x+100000.

$$4\left(x+100000\right)\left(x+100000\right)=11195\left(x+102500\right)x$$

Multiply x+100000 and x+100000 to get \left(x+100000\right)^{2}.

$$4\left(x+100000\right)^{2}=11195\left(x+102500\right)x$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+100000\right)^{2}.

$$4\left(x^{2}+200000x+10000000000\right)=11195\left(x+102500\right)x$$

Use the distributive property to multiply 4 by x^{2}+200000x+10000000000.

$$4x^{2}+800000x+40000000000=11195\left(x+102500\right)x$$

Use the distributive property to multiply 11195 by x+102500.

$$4x^{2}+800000x+40000000000=\left(11195x+1147487500\right)x$$

Use the distributive property to multiply 11195x+1147487500 by x.

$$4x^{2}+800000x+40000000000=11195x^{2}[phone]x$$

Subtract 11195x^{2} from both sides.

$$4x^{2}+800000x+40000000000-11195x^{2}=1147487500x$$

Combine 4x^{2} and -11195x^{2} to get -11191x^{2}.

$$-11191x^{2}+800000x+40000000000=1147487500x$$

Subtract 1147487500x from both sides.

$$-11191x^{2}+800000x+40000000000-1147487500x=0$$

Combine 800000x and -1147487500x to get -1146687500x.

$$-11191x^{2}-1146687500x+40000000000=0$$

Subtract 40000000000 from both sides. Anything subtracted from zero gives its negation.

$$-11191x^{2}-1146687500x=-40000000000$$

Divide both sides by -11191.

$$\frac{-11191x^{2}-1146687500x}{-11191}=-\frac{40000000000}{-11191}$$

Dividing by -11191 undoes the multiplication by -11191.

$$x^{2}+\left(-\frac{1146687500}{-11191}\right)x=-\frac{40000000000}{-11191}$$

Divide -1146687500 by -11191.

$$x^{2}+\frac{1146687500}{11191}x=-\frac{40000000000}{-11191}$$

Divide -40000000000 by -11191.

$$x^{2}+\frac{1146687500}{11191}x=\frac{40000000000}{11191}$$

Divide \frac{1146687500}{11191}, the coefficient of the x term, by 2 to get \frac{573343750}{11191}. Then add the square of \frac{573343750}{11191} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1146687500}{11191}x+\left(\frac{573343750}{11191}\right)^{2}=\frac{40000000000}{11191}+\left(\frac{573343750}{11191}\right)^{2}$$

Square \frac{573343750}{11191} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1146687500}{11191}x+\frac{328723055664062500}{125238481}=\frac{40000000000}{11191}+\frac{328723055664062500}{125238481}$$

Add \frac{40000000000}{11191} to \frac{328723055664062500}{125238481} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1146687500}{11191}x+\frac{328723055664062500}{125238481}=\frac{329170695664062500}{125238481}$$

Factor x^{2}+\frac{1146687500}{11191}x+\frac{328723055664062500}{125238481}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{573343750}{11191}\right)^{2}=\frac{329170695664062500}{125238481}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{573343750}{11191}\right)^{2}}=\sqrt{\frac{329170695664062500}{125238481}}$$

Simplify.

$$x+\frac{573343750}{11191}=\frac{6250\sqrt{8426769809}}{11191} x+\frac{573343750}{11191}=-\frac{6250\sqrt{8426769809}}{11191}$$

Subtract \frac{573343750}{11191} from both sides of the equation.

$$x=\frac{6250\sqrt{8426769809}-573343750}{11191} x=\frac{-6250\sqrt{8426769809}-573343750}{11191}$$