Solve 3v^2/2+60v-4v^2+v^2/2=432

$$\frac { 3 v ^ { 2 } } { 2 } + 60 v - 4 v ^ { 2 } + \frac { v ^ { 2 } } { 2 } = 432$$

Solve for v

$$v=12$$

$$v=18$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$3v^{2}+120v-8v^{2}+v^{2}=864$$

Combine 3v^{2} and -8v^{2} to get -5v^{2}.

$$-5v^{2}+120v+v^{2}=864$$

Combine -5v^{2} and v^{2} to get -4v^{2}.

$$-4v^{2}+120v=864$$

Subtract 864 from both sides.

$$-4v^{2}+120v-864=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 120 for b, and -864 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-120±\sqrt{120^{2}-4\left(-4\right)\left(-864\right)}}{2\left(-4\right)}$$

Square 120.

$$v=\frac{-120±\sqrt{14400-4\left(-4\right)\left(-864\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$v=\frac{-120±\sqrt{14400+16\left(-864\right)}}{2\left(-4\right)}$$

Multiply 16 times -864.

$$v=\frac{-120±\sqrt{14400-13824}}{2\left(-4\right)}$$

Add 14400 to -13824.

$$v=\frac{-120±\sqrt{576}}{2\left(-4\right)}$$

Take the square root of 576.

$$v=\frac{-120±24}{2\left(-4\right)}$$

Multiply 2 times -4.

$$v=\frac{-120±24}{-8}$$

Now solve the equation v=\frac{-120±24}{-8} when ± is plus. Add -120 to 24.

$$v=-\frac{96}{-8}$$

Divide -96 by -8.

$$v=12$$

Now solve the equation v=\frac{-120±24}{-8} when ± is minus. Subtract 24 from -120.

$$v=-\frac{144}{-8}$$

Divide -144 by -8.

$$v=18$$

The equation is now solved.

$$v=12 v=18$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$3v^{2}+120v-8v^{2}+v^{2}=864$$

Combine 3v^{2} and -8v^{2} to get -5v^{2}.

$$-5v^{2}+120v+v^{2}=864$$

Combine -5v^{2} and v^{2} to get -4v^{2}.

$$-4v^{2}+120v=864$$

Divide both sides by -4.

$$\frac{-4v^{2}+120v}{-4}=\frac{864}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$v^{2}+\frac{120}{-4}v=\frac{864}{-4}$$

Divide 120 by -4.

$$v^{2}-30v=\frac{864}{-4}$$

Divide 864 by -4.

$$v^{2}-30v=-216$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}-30v+\left(-15\right)^{2}=-216+\left(-15\right)^{2}$$

Square -15.

$$v^{2}-30v+225=-216+225$$

Add -216 to 225.

$$v^{2}-30v+225=9$$

Factor v^{2}-30v+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v-15\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-15\right)^{2}}=\sqrt{9}$$

Simplify.

$$v-15=3 v-15=-3$$

Add 15 to both sides of the equation.

$$v=18 v=12$$