Solve 30/v_{1}+0.5=45/v_{1+3}

$$\frac { 30 } { v _ { 1 } } + 0.5 = \frac { 45 } { v _ { 1 } + 3 }$$

Solve for v_1

$$v_{1}=12$$

$$v_{1}=15$$

Steps Using the Quadratic Formula

Variable v_{1} cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by v_{1}\left(v_{1}+3\right), the least common multiple of v_{1},v_{1}+3.

$$\left(v_{1}+3\right)\times 30+v_{1}\left(v_{1}+3\right)\times 0.5=v_{1}\times 45$$

Use the distributive property to multiply v_{1}+3 by 30.

$$30v_{1}+90+v_{1}\left(v_{1}+3\right)\times 0.5=v_{1}\times 45$$

Use the distributive property to multiply v_{1} by v_{1}+3.

$$30v_{1}+90+\left(v_{1}^{2}+3v_{1}\right)\times 0.5=v_{1}\times 45$$

Use the distributive property to multiply v_{1}^{2}+3v_{1} by 0.5.

$$30v_{1}+90+0.5v_{1}^{2}+1.5v_{1}=v_{1}\times 45$$

Combine 30v_{1} and 1.5v_{1} to get 31.5v_{1}.

$$31.5v_{1}+90+0.5v_{1}^{2}=v_{1}\times 45$$

Subtract v_{1}\times 45 from both sides.

$$31.5v_{1}+90+0.5v_{1}^{2}-v_{1}\times 45=0$$

Combine 31.5v_{1} and -v_{1}\times 45 to get -13.5v_{1}.

$$-13.5v_{1}+90+0.5v_{1}^{2}=0$$

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.

$$0.5v_{1}^{2}-13.5v_{1}+90=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.5 for a, -13.5 for b, and 90 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v_{1}=\frac{-\left(-13.5\right)±\sqrt{\left(-13.5\right)^{2}-4\times 0.5\times 90}}{2\times 0.5}$$

Square -13.5 by squaring both the numerator and the denominator of the fraction.

$$v_{1}=\frac{-\left(-13.5\right)±\sqrt{182.25-4\times 0.5\times 90}}{2\times 0.5}$$

Multiply -4 times 0.5.

$$v_{1}=\frac{-\left(-13.5\right)±\sqrt{182.25-2\times 90}}{2\times 0.5}$$

Multiply -2 times 90.

$$v_{1}=\frac{-\left(-13.5\right)±\sqrt{182.25-180}}{2\times 0.5}$$

Add 182.25 to -180.

$$v_{1}=\frac{-\left(-13.5\right)±\sqrt{2.25}}{2\times 0.5}$$

Take the square root of 2.25.

$$v_{1}=\frac{-\left(-13.5\right)±\frac{3}{2}}{2\times 0.5}$$

The opposite of -13.5 is 13.5.

$$v_{1}=\frac{13.5±\frac{3}{2}}{2\times 0.5}$$

Multiply 2 times 0.5.

$$v_{1}=\frac{13.5±\frac{3}{2}}{1}$$

Now solve the equation v_{1}=\frac{13.5±\frac{3}{2}}{1} when ± is plus. Add 13.5 to \frac{3}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$v_{1}=\frac{15}{1}$$

Divide 15 by 1.

$$v_{1}=15$$

Now solve the equation v_{1}=\frac{13.5±\frac{3}{2}}{1} when ± is minus. Subtract \frac{3}{2} from 13.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$v_{1}=\frac{12}{1}$$

Divide 12 by 1.

$$v_{1}=12$$

The equation is now solved.

$$v_{1}=15 v_{1}=12$$

Steps for Completing the Square

Variable v_{1} cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by v_{1}\left(v_{1}+3\right), the least common multiple of v_{1},v_{1}+3.

$$\left(v_{1}+3\right)\times 30+v_{1}\left(v_{1}+3\right)\times 0.5=v_{1}\times 45$$

Use the distributive property to multiply v_{1}+3 by 30.

$$30v_{1}+90+v_{1}\left(v_{1}+3\right)\times 0.5=v_{1}\times 45$$

Use the distributive property to multiply v_{1} by v_{1}+3.

$$30v_{1}+90+\left(v_{1}^{2}+3v_{1}\right)\times 0.5=v_{1}\times 45$$

Use the distributive property to multiply v_{1}^{2}+3v_{1} by 0.5.

$$30v_{1}+90+0.5v_{1}^{2}+1.5v_{1}=v_{1}\times 45$$

Combine 30v_{1} and 1.5v_{1} to get 31.5v_{1}.

$$31.5v_{1}+90+0.5v_{1}^{2}=v_{1}\times 45$$

Subtract v_{1}\times 45 from both sides.

$$31.5v_{1}+90+0.5v_{1}^{2}-v_{1}\times 45=0$$

Combine 31.5v_{1} and -v_{1}\times 45 to get -13.5v_{1}.

$$-13.5v_{1}+90+0.5v_{1}^{2}=0$$

Subtract 90 from both sides. Anything subtracted from zero gives its negation.

$$-13.5v_{1}+0.5v_{1}^{2}=-90$$

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.

$$0.5v_{1}^{2}-13.5v_{1}=-90$$

Multiply both sides by 2.

$$\frac{0.5v_{1}^{2}-13.5v_{1}}{0.5}=-\frac{90}{0.5}$$

Dividing by 0.5 undoes the multiplication by 0.5.

$$v_{1}^{2}+\left(-\frac{13.5}{0.5}\right)v_{1}=-\frac{90}{0.5}$$

Divide -13.5 by 0.5 by multiplying -13.5 by the reciprocal of 0.5.

$$v_{1}^{2}-27v_{1}=-\frac{90}{0.5}$$

Divide -90 by 0.5 by multiplying -90 by the reciprocal of 0.5.

$$v_{1}^{2}-27v_{1}=-180$$

Divide -27, the coefficient of the x term, by 2 to get -\frac{27}{2}. Then add the square of -\frac{27}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v_{1}^{2}-27v_{1}+\left(-\frac{27}{2}\right)^{2}=-180+\left(-\frac{27}{2}\right)^{2}$$

Square -\frac{27}{2} by squaring both the numerator and the denominator of the fraction.

$$v_{1}^{2}-27v_{1}+182.25=-180+182.25$$

Add -180 to 182.25.

$$v_{1}^{2}-27v_{1}+182.25=2.25$$

Factor v_{1}^{2}-27v_{1}+182.25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v_{1}-\frac{27}{2}\right)^{2}=2.25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v_{1}-\frac{27}{2}\right)^{2}}=\sqrt{2.25}$$

Simplify.

$$v_{1}-\frac{27}{2}=\frac{3}{2} v_{1}-\frac{27}{2}=-\frac{3}{2}$$

Add \frac{27}{2} to both sides of the equation.

$$v_{1}=15 v_{1}=12$$