Solve (6v-9)(2v+1)=7v^2-38-33

$$( 6 v - 9 ) ( 2 v + 1 ) = 7 v ^ { 2 } - 38 - 33$$

Solve for v

$$v=\frac{6+\sqrt{274}i}{5}\approx 1.2+3.310589071i$$

$$v=\frac{-\sqrt{274}i+6}{5}\approx 1.2-3.310589071i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6v-9 by 2v+1 and combine like terms.

$$12v^{2}-12v-9=7v^{2}-38-33$$

Subtract 33 from -38 to get -71.

$$12v^{2}-12v-9=7v^{2}-71$$

Subtract 7v^{2} from both sides.

$$12v^{2}-12v-9-7v^{2}=-71$$

Combine 12v^{2} and -7v^{2} to get 5v^{2}.

$$5v^{2}-12v-9=-71$$

Add 71 to both sides.

$$5v^{2}-12v-9+71=0$$

Add -9 and 71 to get 62.

$$5v^{2}-12v+62=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -12 for b, and 62 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 5\times 62}}{2\times 5}$$

Square -12.

$$v=\frac{-\left(-12\right)±\sqrt{144-4\times 5\times 62}}{2\times 5}$$

Multiply -4 times 5.

$$v=\frac{-\left(-12\right)±\sqrt{144-20\times 62}}{2\times 5}$$

Multiply -20 times 62.

$$v=\frac{-\left(-12\right)±\sqrt{144-1240}}{2\times 5}$$

Add 144 to -1240.

$$v=\frac{-\left(-12\right)±\sqrt{-1096}}{2\times 5}$$

Take the square root of -1096.

$$v=\frac{-\left(-12\right)±2\sqrt{274}i}{2\times 5}$$

The opposite of -12 is 12.

$$v=\frac{12±2\sqrt{274}i}{2\times 5}$$

Multiply 2 times 5.

$$v=\frac{12±2\sqrt{274}i}{10}$$

Now solve the equation v=\frac{12±2\sqrt{274}i}{10} when ± is plus. Add 12 to 2i\sqrt{274}.

$$v=\frac{12+2\sqrt{274}i}{10}$$

Divide 12+2i\sqrt{274} by 10.

$$v=\frac{6+\sqrt{274}i}{5}$$

Now solve the equation v=\frac{12±2\sqrt{274}i}{10} when ± is minus. Subtract 2i\sqrt{274} from 12.

$$v=\frac{-2\sqrt{274}i+12}{10}$$

Divide 12-2i\sqrt{274} by 10.

$$v=\frac{-\sqrt{274}i+6}{5}$$

The equation is now solved.

$$v=\frac{6+\sqrt{274}i}{5} v=\frac{-\sqrt{274}i+6}{5}$$

Steps for Completing the Square

Use the distributive property to multiply 6v-9 by 2v+1 and combine like terms.

$$12v^{2}-12v-9=7v^{2}-38-33$$

Subtract 33 from -38 to get -71.

$$12v^{2}-12v-9=7v^{2}-71$$

Subtract 7v^{2} from both sides.

$$12v^{2}-12v-9-7v^{2}=-71$$

Combine 12v^{2} and -7v^{2} to get 5v^{2}.

$$5v^{2}-12v-9=-71$$

Add 9 to both sides.

$$5v^{2}-12v=-71+9$$

Add -71 and 9 to get -62.

$$5v^{2}-12v=-62$$

Divide both sides by 5.

$$\frac{5v^{2}-12v}{5}=-\frac{62}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$v^{2}-\frac{12}{5}v=-\frac{62}{5}$$

Divide -\frac{12}{5}, the coefficient of the x term, by 2 to get -\frac{6}{5}. Then add the square of -\frac{6}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}-\frac{12}{5}v+\left(-\frac{6}{5}\right)^{2}=-\frac{62}{5}+\left(-\frac{6}{5}\right)^{2}$$

Square -\frac{6}{5} by squaring both the numerator and the denominator of the fraction.

$$v^{2}-\frac{12}{5}v+\frac{36}{25}=-\frac{62}{5}+\frac{36}{25}$$

Add -\frac{62}{5} to \frac{36}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$v^{2}-\frac{12}{5}v+\frac{36}{25}=-\frac{274}{25}$$

Factor v^{2}-\frac{12}{5}v+\frac{36}{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}.

$$\left(v-\frac{6}{5}\right)^{2}=-\frac{274}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-\frac{6}{5}\right)^{2}}=\sqrt{-\frac{274}{25}}$$

Simplify.

$$v-\frac{6}{5}=\frac{\sqrt{274}i}{5} v-\frac{6}{5}=-\frac{\sqrt{274}i}{5}$$

Add \frac{6}{5} to both sides of the equation.

$$v=\frac{6+\sqrt{274}i}{5} v=\frac{-\sqrt{274}i+6}{5}$$