Solve v^2=4v-13

$$v ^ { 2 } = 4 v - 13$$

Solve for v

$$v=2+3i$$

$$v=2-3i$$

Steps Using the Quadratic Formula

Subtract 4v from both sides.

$$v^{2}-4v=-13$$

Add 13 to both sides.

$$v^{2}-4v+13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and 13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 13}}{2}$$

Square -4.

$$v=\frac{-\left(-4\right)±\sqrt{16-4\times 13}}{2}$$

Multiply -4 times 13.

$$v=\frac{-\left(-4\right)±\sqrt{16-52}}{2}$$

Add 16 to -52.

$$v=\frac{-\left(-4\right)±\sqrt{-36}}{2}$$

Take the square root of -36.

$$v=\frac{-\left(-4\right)±6i}{2}$$

The opposite of -4 is 4.

$$v=\frac{4±6i}{2}$$

Now solve the equation v=\frac{4±6i}{2} when ± is plus. Add 4 to 6i.

$$v=\frac{4+6i}{2}$$

Divide 4+6i by 2.

$$v=2+3i$$

Now solve the equation v=\frac{4±6i}{2} when ± is minus. Subtract 6i from 4.

$$v=\frac{4-6i}{2}$$

Divide 4-6i by 2.

$$v=2-3i$$

The equation is now solved.

$$v=2+3i v=2-3i$$

Steps for Completing the Square

Subtract 4v from both sides.

$$v^{2}-4v=-13$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}-4v+\left(-2\right)^{2}=-13+\left(-2\right)^{2}$$

Square -2.

$$v^{2}-4v+4=-13+4$$

Add -13 to 4.

$$v^{2}-4v+4=-9$$

Factor v^{2}-4v+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$v-2=3i v-2=-3i$$

Add 2 to both sides of the equation.

$$v=2+3i v=2-3i$$