Solve 2450=v^2+28v

$$2450 = v ^ { 2 } + 28 v$$

Solve for v

$$v=21\sqrt{6}-14\approx 37.439284598$$

$$v=-21\sqrt{6}-14\approx -65.439284598$$

Steps Using the Quadratic Formula

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

$$v^{2}+28v=2450$$

Subtract 2450 from both sides.

$$v^{2}+28v-2450=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 28 for b, and -2450 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-28±\sqrt{28^{2}-4\left(-2450\right)}}{2}$$

Square 28.

$$v=\frac{-28±\sqrt{784-4\left(-2450\right)}}{2}$$

Multiply -4 times -2450.

$$v=\frac{-28±\sqrt{784+9800}}{2}$$

Add 784 to 9800.

$$v=\frac{-28±\sqrt{10584}}{2}$$

Take the square root of 10584.

$$v=\frac{-28±42\sqrt{6}}{2}$$

Now solve the equation v=\frac{-28±42\sqrt{6}}{2} when ± is plus. Add -28 to 42\sqrt{6}.

$$v=\frac{42\sqrt{6}-28}{2}$$

Divide -28+42\sqrt{6} by 2.

$$v=21\sqrt{6}-14$$

Now solve the equation v=\frac{-28±42\sqrt{6}}{2} when ± is minus. Subtract 42\sqrt{6} from -28.

$$v=\frac{-42\sqrt{6}-28}{2}$$

Divide -28-42\sqrt{6} by 2.

$$v=-21\sqrt{6}-14$$

The equation is now solved.

$$v=21\sqrt{6}-14 v=-21\sqrt{6}-14$$

Steps for Completing the Square

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

$$v^{2}+28v=2450$$

Divide 28, the coefficient of the x term, by 2 to get 14. Then add the square of 14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+28v+14^{2}=2450+14^{2}$$

Square 14.

$$v^{2}+28v+196=2450+196$$

Add 2450 to 196.

$$v^{2}+28v+196=2646$$

Factor v^{2}+28v+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v+14\right)^{2}=2646$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+14\right)^{2}}=\sqrt{2646}$$

Simplify.

$$v+14=21\sqrt{6} v+14=-21\sqrt{6}$$

Subtract 14 from both sides of the equation.

$$v=21\sqrt{6}-14 v=-21\sqrt{6}-14$$