Solve 99/v+10=90/v-3/4

$$\frac { 99 } { v + 10 } = \frac { 90 } { v } - \frac { 3 } { 4 }$$

Solve for v (complex solution)

$$v=\sqrt{1321}-11\approx 25.345563691$$

$$v=-\left(\sqrt{1321}+11\right)\approx -47.345563691$$

Steps Using the Quadratic Formula

Variable v cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 4v\left(v+10\right), the least common multiple of v+10,v,4.

$$4v\times 99=\left(4v+40\right)\times 90+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

Multiply 4 and 99 to get 396.

$$396v=\left(4v+40\right)\times 90+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

Use the distributive property to multiply 4v+40 by 90.

$$396v=360v+3600+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

Multiply 4 and -\frac{3}{4} to get -3.

$$396v=360v+3600-3v\left(v+10\right)$$

Use the distributive property to multiply -3v by v+10.

$$396v=360v+3600-3v^{2}-30v$$

Combine 360v and -30v to get 330v.

$$396v=330v+3600-3v^{2}$$

Subtract 330v from both sides.

$$396v-330v=3600-3v^{2}$$

Combine 396v and -330v to get 66v.

$$66v=3600-3v^{2}$$

Subtract 3600 from both sides.

$$66v-3600=-3v^{2}$$

Add 3v^{2} to both sides.

$$66v-3600+3v^{2}=0$$

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.

$$3v^{2}+66v-3600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 66 for b, and -3600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-66±\sqrt{66^{2}-4\times 3\left(-3600\right)}}{2\times 3}$$

Square 66.

$$v=\frac{-66±\sqrt{4356-4\times 3\left(-3600\right)}}{2\times 3}$$

Multiply -4 times 3.

$$v=\frac{-66±\sqrt{4356-12\left(-3600\right)}}{2\times 3}$$

Multiply -12 times -3600.

$$v=\frac{-66±\sqrt{4356+43200}}{2\times 3}$$

Add 4356 to 43200.

$$v=\frac{-66±\sqrt{47556}}{2\times 3}$$

Take the square root of 47556.

$$v=\frac{-66±6\sqrt{1321}}{2\times 3}$$

Multiply 2 times 3.

$$v=\frac{-66±6\sqrt{1321}}{6}$$

Now solve the equation v=\frac{-66±6\sqrt{1321}}{6} when ± is plus. Add -66 to 6\sqrt{1321}.

$$v=\frac{6\sqrt{1321}-66}{6}$$

Divide -66+6\sqrt{1321} by 6.

$$v=\sqrt{1321}-11$$

Now solve the equation v=\frac{-66±6\sqrt{1321}}{6} when ± is minus. Subtract 6\sqrt{1321} from -66.

$$v=\frac{-6\sqrt{1321}-66}{6}$$

Divide -66-6\sqrt{1321} by 6.

$$v=-\sqrt{1321}-11$$

The equation is now solved.

$$v=\sqrt{1321}-11 v=-\sqrt{1321}-11$$

Steps for Completing the Square

Variable v cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 4v\left(v+10\right), the least common multiple of v+10,v,4.

$$4v\times 99=\left(4v+40\right)\times 90+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

Multiply 4 and 99 to get 396.

$$396v=\left(4v+40\right)\times 90+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

Use the distributive property to multiply 4v+40 by 90.

$$396v=360v+3600+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

Multiply 4 and -\frac{3}{4} to get -3.

$$396v=360v+3600-3v\left(v+10\right)$$

Use the distributive property to multiply -3v by v+10.

$$396v=360v+3600-3v^{2}-30v$$

Combine 360v and -30v to get 330v.

$$396v=330v+3600-3v^{2}$$

Subtract 330v from both sides.

$$396v-330v=3600-3v^{2}$$

Combine 396v and -330v to get 66v.

$$66v=3600-3v^{2}$$

Add 3v^{2} to both sides.

$$66v+3v^{2}=3600$$

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.

$$3v^{2}+66v=3600$$

Divide both sides by 3.

$$\frac{3v^{2}+66v}{3}=\frac{3600}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$v^{2}+\frac{66}{3}v=\frac{3600}{3}$$

Divide 66 by 3.

$$v^{2}+22v=\frac{3600}{3}$$

Divide 3600 by 3.

$$v^{2}+22v=1200$$

Divide 22, the coefficient of the x term, by 2 to get 11. Then add the square of 11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+22v+11^{2}=1200+11^{2}$$

Square 11.

$$v^{2}+22v+121=1200+121$$

Add 1200 to 121.

$$v^{2}+22v+121=1321$$

Factor v^{2}+22v+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v+11\right)^{2}=1321$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+11\right)^{2}}=\sqrt{1321}$$

Simplify.

$$v+11=\sqrt{1321} v+11=-\sqrt{1321}$$

Subtract 11 from both sides of the equation.

$$v=\sqrt{1321}-11 v=-\sqrt{1321}-11$$

Solve for v

$$v=\sqrt{1321}-11\approx 25.345563691$$

$$v=-\sqrt{1321}-11\approx -47.345563691$$

Steps Using the Quadratic Formula

Variable v cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 4v\left(v+10\right), the least common multiple of v+10,v,4.

$$4v\times 99=\left(4v+40\right)\times 90+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

Multiply 4 and 99 to get 396.

$$396v=\left(4v+40\right)\times 90+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

Use the distributive property to multiply 4v+40 by 90.

$$396v=360v+3600+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

Multiply 4 and -\frac{3}{4} to get -3.

$$396v=360v+3600-3v\left(v+10\right)$$

Use the distributive property to multiply -3v by v+10.

$$396v=360v+3600-3v^{2}-30v$$

Combine 360v and -30v to get 330v.

$$396v=330v+3600-3v^{2}$$

Subtract 330v from both sides.

$$396v-330v=3600-3v^{2}$$

Combine 396v and -330v to get 66v.

$$66v=3600-3v^{2}$$

Subtract 3600 from both sides.

$$66v-3600=-3v^{2}$$

Add 3v^{2} to both sides.

$$66v-3600+3v^{2}=0$$

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.

$$3v^{2}+66v-3600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 66 for b, and -3600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-66±\sqrt{66^{2}-4\times 3\left(-3600\right)}}{2\times 3}$$

Square 66.

$$v=\frac{-66±\sqrt{4356-4\times 3\left(-3600\right)}}{2\times 3}$$

Multiply -4 times 3.

$$v=\frac{-66±\sqrt{4356-12\left(-3600\right)}}{2\times 3}$$

Multiply -12 times -3600.

$$v=\frac{-66±\sqrt{4356+43200}}{2\times 3}$$

Add 4356 to 43200.

$$v=\frac{-66±\sqrt{47556}}{2\times 3}$$

Take the square root of 47556.

$$v=\frac{-66±6\sqrt{1321}}{2\times 3}$$

Multiply 2 times 3.

$$v=\frac{-66±6\sqrt{1321}}{6}$$

Now solve the equation v=\frac{-66±6\sqrt{1321}}{6} when ± is plus. Add -66 to 6\sqrt{1321}.

$$v=\frac{6\sqrt{1321}-66}{6}$$

Divide -66+6\sqrt{1321} by 6.

$$v=\sqrt{1321}-11$$

Now solve the equation v=\frac{-66±6\sqrt{1321}}{6} when ± is minus. Subtract 6\sqrt{1321} from -66.

$$v=\frac{-6\sqrt{1321}-66}{6}$$

Divide -66-6\sqrt{1321} by 6.

$$v=-\sqrt{1321}-11$$

The equation is now solved.

$$v=\sqrt{1321}-11 v=-\sqrt{1321}-11$$

Steps for Completing the Square

Variable v cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 4v\left(v+10\right), the least common multiple of v+10,v,4.

$$4v\times 99=\left(4v+40\right)\times 90+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

Multiply 4 and 99 to get 396.

$$396v=\left(4v+40\right)\times 90+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

Use the distributive property to multiply 4v+40 by 90.

$$396v=360v+3600+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

Multiply 4 and -\frac{3}{4} to get -3.

$$396v=360v+3600-3v\left(v+10\right)$$

Use the distributive property to multiply -3v by v+10.

$$396v=360v+3600-3v^{2}-30v$$

Combine 360v and -30v to get 330v.

$$396v=330v+3600-3v^{2}$$

Subtract 330v from both sides.

$$396v-330v=3600-3v^{2}$$

Combine 396v and -330v to get 66v.

$$66v=3600-3v^{2}$$

Add 3v^{2} to both sides.

$$66v+3v^{2}=3600$$

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.

$$3v^{2}+66v=3600$$

Divide both sides by 3.

$$\frac{3v^{2}+66v}{3}=\frac{3600}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$v^{2}+\frac{66}{3}v=\frac{3600}{3}$$

Divide 66 by 3.

$$v^{2}+22v=\frac{3600}{3}$$

Divide 3600 by 3.

$$v^{2}+22v=1200$$

Divide 22, the coefficient of the x term, by 2 to get 11. Then add the square of 11 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+22v+11^{2}=1200+11^{2}$$

Square 11.

$$v^{2}+22v+121=1200+121$$

Add 1200 to 121.

$$v^{2}+22v+121=1321$$

Factor v^{2}+22v+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v+11\right)^{2}=1321$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+11\right)^{2}}=\sqrt{1321}$$

Simplify.

$$v+11=\sqrt{1321} v+11=-\sqrt{1321}$$

Subtract 11 from both sides of the equation.

$$v=\sqrt{1321}-11 v=-\sqrt{1321}-11$$