Solve 104=v(v-3)/2

$$104 = \frac { v ( v - 3 ) } { 2 }$$

Solve for v

$$v=-13$$

$$v=16$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$104\times 2=v\left(v-3\right)$$

Multiply 104 and 2 to get 208.

$$208=v\left(v-3\right)$$

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

$$208=v^{2}-3v$$

Swap sides so that all variable terms are on the left hand side.

$$v^{2}-3v=208$$

Subtract 208 from both sides.

$$v^{2}-3v-208=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -208 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-208\right)}}{2}$$

Square -3.

$$v=\frac{-\left(-3\right)±\sqrt{9-4\left(-208\right)}}{2}$$

Multiply -4 times -208.

$$v=\frac{-\left(-3\right)±\sqrt{9+832}}{2}$$

Add 9 to 832.

$$v=\frac{-\left(-3\right)±\sqrt{841}}{2}$$

Take the square root of 841.

$$v=\frac{-\left(-3\right)±29}{2}$$

The opposite of -3 is 3.

$$v=\frac{3±29}{2}$$

Now solve the equation v=\frac{3±29}{2} when ± is plus. Add 3 to 29.

$$v=\frac{32}{2}$$

Divide 32 by 2.

$$v=16$$

Now solve the equation v=\frac{3±29}{2} when ± is minus. Subtract 29 from 3.

$$v=-\frac{26}{2}$$

Divide -26 by 2.

$$v=-13$$

The equation is now solved.

$$v=16 v=-13$$

Steps for Completing the Square

Multiply both sides by 2.

$$104\times 2=v\left(v-3\right)$$

Multiply 104 and 2 to get 208.

$$208=v\left(v-3\right)$$

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

$$208=v^{2}-3v$$

Swap sides so that all variable terms are on the left hand side.

$$v^{2}-3v=208$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}-3v+\left(-\frac{3}{2}\right)^{2}=208+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$v^{2}-3v+\frac{9}{4}=208+\frac{9}{4}$$

Add 208 to \frac{9}{4}.

$$v^{2}-3v+\frac{9}{4}=\frac{841}{4}$$

Factor v^{2}-3v+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v-\frac{3}{2}\right)^{2}=\frac{841}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-\frac{3}{2}\right)^{2}}=\sqrt{\frac{841}{4}}$$

Simplify.

$$v-\frac{3}{2}=\frac{29}{2} v-\frac{3}{2}=-\frac{29}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$v=16 v=-13$$