Solve 25/x=30/11+20000x+10/60

$$\frac { 25 } { x } = \frac { 30 } { 11 + 20000 x } + \frac { 10 } { 60 }$$

Solve for x

$$x = \frac{\sqrt{8998986036481} + 2999809}{40000} \approx 149.991000033$$

$$x=\frac{2999809-\sqrt{8998986036481}}{40000}\approx -0.000550033$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{11}{20000},0 since division by zero is not defined. Multiply both sides of the equation by 60x\left(20000x+11\right), the least common multiple of x,11+20000x,60.

$$\left(1200000x+660\right)\times 25=60x\times 30+x\left(20000x+11\right)\times 10$$

Use the distributive property to multiply 1200000x+660 by 25.

$$30000000x+16500=60x\times 30+x\left(20000x+11\right)\times 10$$

Multiply 60 and 30 to get 1800.

$$30000000x+16500=1800x+x\left(20000x+11\right)\times 10$$

Use the distributive property to multiply x by 20000x+11.

$$30000000x+16500=1800x+\left(20000x^{2}+11x\right)\times 10$$

Use the distributive property to multiply 20000x^{2}+11x by 10.

$$30000000x+16500=1800x+200000x^{2}+110x$$

Combine 1800x and 110x to get 1910x.

$$30000000x+16500=1910x+200000x^{2}$$

Subtract 1910x from both sides.

$$30000000x+16500-1910x=200000x^{2}$$

Combine 30000000x and -1910x to get 29998090x.

$$29998090x+16500=200000x^{2}$$

Subtract 200000x^{2} from both sides.

$$29998090x+16500-200000x^{2}=0$$

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.

$$-200000x^{2}[phone]x+16500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -200000 for a, 29998090 for b, and 16500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-29998090±\sqrt{29998090^{2}-4\left(-200000\right)\times 16500}}{2\left(-200000\right)}$$

Square 29998090.

$$x=\frac{-29998090±\sqrt{899885403648100-4\left(-200000\right)\times 16500}}{2\left(-200000\right)}$$

Multiply -4 times -200000.

$$x=\frac{-29998090±\sqrt{899885403648100+800000\times 16500}}{2\left(-200000\right)}$$

Multiply 800000 times 16500.

$$x=\frac{-29998090±\sqrt{899885403648100+13200000000}}{2\left(-200000\right)}$$

Add 899885403648100 to 13200000000.

$$x=\frac{-29998090±\sqrt{899898603648100}}{2\left(-200000\right)}$$

Take the square root of 899898603648100.

$$x=\frac{-29998090±10\sqrt{8998986036481}}{2\left(-200000\right)}$$

Multiply 2 times -200000.

$$x=\frac{-29998090±10\sqrt{8998986036481}}{-400000}$$

Now solve the equation x=\frac{-29998090±10\sqrt{8998986036481}}{-400000} when ± is plus. Add -29998090 to 10\sqrt{8998986036481}.

$$x=\frac{10\sqrt{8998986036481}-29998090}{-400000}$$

Divide -29998090+10\sqrt{8998986036481} by -400000.

$$x=\frac{2999809-\sqrt{8998986036481}}{40000}$$

Now solve the equation x=\frac{-29998090±10\sqrt{8998986036481}}{-400000} when ± is minus. Subtract 10\sqrt{8998986036481} from -29998090.

$$x=\frac{-10\sqrt{8998986036481}-29998090}{-400000}$$

Divide -29998090-10\sqrt{8998986036481} by -400000.

$$x=\frac{\sqrt{8998986036481}+2999809}{40000}$$

The equation is now solved.

$$x=\frac{2999809-\sqrt{8998986036481}}{40000} x=\frac{\sqrt{8998986036481}+2999809}{40000}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{11}{20000},0 since division by zero is not defined. Multiply both sides of the equation by 60x\left(20000x+11\right), the least common multiple of x,11+20000x,60.

$$\left(1200000x+660\right)\times 25=60x\times 30+x\left(20000x+11\right)\times 10$$

Use the distributive property to multiply 1200000x+660 by 25.

$$30000000x+16500=60x\times 30+x\left(20000x+11\right)\times 10$$

Multiply 60 and 30 to get 1800.

$$30000000x+16500=1800x+x\left(20000x+11\right)\times 10$$

Use the distributive property to multiply x by 20000x+11.

$$30000000x+16500=1800x+\left(20000x^{2}+11x\right)\times 10$$

Use the distributive property to multiply 20000x^{2}+11x by 10.

$$30000000x+16500=1800x+200000x^{2}+110x$$

Combine 1800x and 110x to get 1910x.

$$30000000x+16500=1910x+200000x^{2}$$

Subtract 1910x from both sides.

$$30000000x+16500-1910x=200000x^{2}$$

Combine 30000000x and -1910x to get 29998090x.

$$29998090x+16500=200000x^{2}$$

Subtract 200000x^{2} from both sides.

$$29998090x+16500-200000x^{2}=0$$

Subtract 16500 from both sides. Anything subtracted from zero gives its negation.

$$29998090x-200000x^{2}=-16500$$

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.

$$-200000x^{2}[phone]x=-16500$$

Divide both sides by -200000.

$$\frac{-200000x^{2}[phone]x}{-200000}=-\frac{16500}{-200000}$$

Dividing by -200000 undoes the multiplication by -200000.

$$x^{2}+\frac{29998090}{-200000}x=-\frac{16500}{-200000}$$

Reduce the fraction \frac{29998090}{-200000} to lowest terms by extracting and canceling out 10.

$$x^{2}-\frac{2999809}{20000}x=-\frac{16500}{-200000}$$

Reduce the fraction \frac{-16500}{-200000} to lowest terms by extracting and canceling out 500.

$$x^{2}-\frac{2999809}{20000}x=\frac{33}{400}$$

Divide -\frac{2999809}{20000}, the coefficient of the x term, by 2 to get -\frac{2999809}{40000}. Then add the square of -\frac{2999809}{40000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2999809}{20000}x+\left(-\frac{2999809}{40000}\right)^{2}=\frac{33}{400}+\left(-\frac{2999809}{40000}\right)^{2}$$

Square -\frac{2999809}{40000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2999809}{20000}x+\frac{8998854036481}{1600000000}=\frac{33}{400}+\frac{8998854036481}{1600000000}$$

Add \frac{33}{400} to \frac{8998854036481}{1600000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2999809}{20000}x+\frac{8998854036481}{1600000000}=\frac{8998986036481}{1600000000}$$

Factor x^{2}-\frac{2999809}{20000}x+\frac{8998854036481}{1600000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2999809}{40000}\right)^{2}=\frac{8998986036481}{1600000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2999809}{40000}\right)^{2}}=\sqrt{\frac{8998986036481}{1600000000}}$$

Simplify.

$$x-\frac{2999809}{40000}=\frac{\sqrt{8998986036481}}{40000} x-\frac{2999809}{40000}=-\frac{\sqrt{8998986036481}}{40000}$$

Add \frac{2999809}{40000} to both sides of the equation.

$$x=\frac{\sqrt{8998986036481}+2999809}{40000} x=\frac{2999809-\sqrt{8998986036481}}{40000}$$