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

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

Solve for v

$$v=-40$$

$$v=30$$

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.

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

Multiply 4 and 90 to get 360.

$$360v=\left(4v+40\right)\times 90+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

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

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

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

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

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

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

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

$$360v=330v+3600-3v^{2}$$

Subtract 330v from both sides.

$$360v-330v=3600-3v^{2}$$

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

$$30v=3600-3v^{2}$$

Subtract 3600 from both sides.

$$30v-3600=-3v^{2}$$

Add 3v^{2} to both sides.

$$30v-3600+3v^{2}=0$$

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.

$$3v^{2}+30v-3600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 30 for b, and -3600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-30±\sqrt{30^{2}-4\times 3\left(-3600\right)}}{2\times 3}$$

Square 30.

$$v=\frac{-30±\sqrt{900-4\times 3\left(-3600\right)}}{2\times 3}$$

Multiply -4 times 3.

$$v=\frac{-30±\sqrt{900-12\left(-3600\right)}}{2\times 3}$$

Multiply -12 times -3600.

$$v=\frac{-30±\sqrt{900+43200}}{2\times 3}$$

Add 900 to 43200.

$$v=\frac{-30±\sqrt{44100}}{2\times 3}$$

Take the square root of 44100.

$$v=\frac{-30±210}{2\times 3}$$

Multiply 2 times 3.

$$v=\frac{-30±210}{6}$$

Now solve the equation v=\frac{-30±210}{6} when ± is plus. Add -30 to 210.

$$v=\frac{180}{6}$$

Divide 180 by 6.

$$v=30$$

Now solve the equation v=\frac{-30±210}{6} when ± is minus. Subtract 210 from -30.

$$v=-\frac{240}{6}$$

Divide -240 by 6.

$$v=-40$$

The equation is now solved.

$$v=30 v=-40$$

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.

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

Multiply 4 and 90 to get 360.

$$360v=\left(4v+40\right)\times 90+4v\left(v+10\right)\left(-\frac{3}{4}\right)$$

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

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

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

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

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

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

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

$$360v=330v+3600-3v^{2}$$

Subtract 330v from both sides.

$$360v-330v=3600-3v^{2}$$

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

$$30v=3600-3v^{2}$$

Add 3v^{2} to both sides.

$$30v+3v^{2}=3600$$

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.

$$3v^{2}+30v=3600$$

Divide both sides by 3.

$$\frac{3v^{2}+30v}{3}=\frac{3600}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$v^{2}+\frac{30}{3}v=\frac{3600}{3}$$

Divide 30 by 3.

$$v^{2}+10v=\frac{3600}{3}$$

Divide 3600 by 3.

$$v^{2}+10v=1200$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+10v+5^{2}=1200+5^{2}$$

Square 5.

$$v^{2}+10v+25=1200+25$$

Add 1200 to 25.

$$v^{2}+10v+25=1225$$

Factor v^{2}+10v+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v+5\right)^{2}=1225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+5\right)^{2}}=\sqrt{1225}$$

Simplify.

$$v+5=35 v+5=-35$$

Subtract 5 from both sides of the equation.

$$v=30 v=-40$$