Solve 7v-4(v-9)=4(5v+9)v

$$7 v - 4 ( v - 9 ) = 4 ( 5 v + 9 ) v$$

Solve for v

$$v=\frac{3}{4}=0.75$$

$$v = -\frac{12}{5} = -2\frac{2}{5} = -2.4$$

Steps Using Factoring By Grouping

Use the distributive property to multiply -4 by v-9.

$$7v-4v+36=4\left(5v+9\right)v$$

Combine 7v and -4v to get 3v.

$$3v+36=4\left(5v+9\right)v$$

Use the distributive property to multiply 4 by 5v+9.

$$3v+36=\left(20v+36\right)v$$

Use the distributive property to multiply 20v+36 by v.

$$3v+36=20v^{2}+36v$$

Subtract 20v^{2} from both sides.

$$3v+36-20v^{2}=36v$$

Subtract 36v from both sides.

$$3v+36-20v^{2}-36v=0$$

Combine 3v and -36v to get -33v.

$$-33v+36-20v^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-20v^{2}-33v+36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -20v^{2}+av+bv+36. To find a and b, set up a system to be solved.

$$a+b=-33 ab=-20\times 36=-720$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -720.

$$1,-720 2,-360 3,-240 4,-180 5,-144 6,-120 8,-90 9,-80 10,-72 12,-60 15,-48 16,-45 18,-40 20,-36 24,-30$$

Calculate the sum for each pair.

$$1-720=-719 2-360=-358 3-240=-237 4-180=-176 5-144=-139 6-120=-114 8-90=-82 9-80=-71 10-72=-62 12-60=-48 15-48=-33 16-45=-29 18-40=-22 20-36=-16 24-30=-6$$

The solution is the pair that gives sum -33.

$$a=15 b=-48$$

Rewrite -20v^{2}-33v+36 as \left(-20v^{2}+15v\right)+\left(-48v+36\right).

$$\left(-20v^{2}+15v\right)+\left(-48v+36\right)$$

Factor out -5v in the first and -12 in the second group.

$$-5v\left(4v-3\right)-12\left(4v-3\right)$$

Factor out common term 4v-3 by using distributive property.

$$\left(4v-3\right)\left(-5v-12\right)$$

To find equation solutions, solve 4v-3=0 and -5v-12=0.

$$v=\frac{3}{4} v=-\frac{12}{5}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -4 by v-9.

$$7v-4v+36=4\left(5v+9\right)v$$

Combine 7v and -4v to get 3v.

$$3v+36=4\left(5v+9\right)v$$

Use the distributive property to multiply 4 by 5v+9.

$$3v+36=\left(20v+36\right)v$$

Use the distributive property to multiply 20v+36 by v.

$$3v+36=20v^{2}+36v$$

Subtract 20v^{2} from both sides.

$$3v+36-20v^{2}=36v$$

Subtract 36v from both sides.

$$3v+36-20v^{2}-36v=0$$

Combine 3v and -36v to get -33v.

$$-33v+36-20v^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-20v^{2}-33v+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, -33 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\left(-20\right)\times 36}}{2\left(-20\right)}$$

Square -33.

$$v=\frac{-\left(-33\right)±\sqrt{1089-4\left(-20\right)\times 36}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$v=\frac{-\left(-33\right)±\sqrt{1089+80\times 36}}{2\left(-20\right)}$$

Multiply 80 times 36.

$$v=\frac{-\left(-33\right)±\sqrt{1089+2880}}{2\left(-20\right)}$$

Add 1089 to 2880.

$$v=\frac{-\left(-33\right)±\sqrt{3969}}{2\left(-20\right)}$$

Take the square root of 3969.

$$v=\frac{-\left(-33\right)±63}{2\left(-20\right)}$$

The opposite of -33 is 33.

$$v=\frac{33±63}{2\left(-20\right)}$$

Multiply 2 times -20.

$$v=\frac{33±63}{-40}$$

Now solve the equation v=\frac{33±63}{-40} when ± is plus. Add 33 to 63.

$$v=\frac{96}{-40}$$

Reduce the fraction \frac{96}{-40} to lowest terms by extracting and canceling out 8.

$$v=-\frac{12}{5}$$

Now solve the equation v=\frac{33±63}{-40} when ± is minus. Subtract 63 from 33.

$$v=-\frac{30}{-40}$$

Reduce the fraction \frac{-30}{-40} to lowest terms by extracting and canceling out 10.

$$v=\frac{3}{4}$$

The equation is now solved.

$$v=-\frac{12}{5} v=\frac{3}{4}$$

Steps for Completing the Square

Use the distributive property to multiply -4 by v-9.

$$7v-4v+36=4\left(5v+9\right)v$$

Combine 7v and -4v to get 3v.

$$3v+36=4\left(5v+9\right)v$$

Use the distributive property to multiply 4 by 5v+9.

$$3v+36=\left(20v+36\right)v$$

Use the distributive property to multiply 20v+36 by v.

$$3v+36=20v^{2}+36v$$

Subtract 20v^{2} from both sides.

$$3v+36-20v^{2}=36v$$

Subtract 36v from both sides.

$$3v+36-20v^{2}-36v=0$$

Combine 3v and -36v to get -33v.

$$-33v+36-20v^{2}=0$$

Subtract 36 from both sides. Anything subtracted from zero gives its negation.

$$-33v-20v^{2}=-36$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-20v^{2}-33v=-36$$

Divide both sides by -20.

$$\frac{-20v^{2}-33v}{-20}=-\frac{36}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$v^{2}+\left(-\frac{33}{-20}\right)v=-\frac{36}{-20}$$

Divide -33 by -20.

$$v^{2}+\frac{33}{20}v=-\frac{36}{-20}$$

Reduce the fraction \frac{-36}{-20} to lowest terms by extracting and canceling out 4.

$$v^{2}+\frac{33}{20}v=\frac{9}{5}$$

Divide \frac{33}{20}, the coefficient of the x term, by 2 to get \frac{33}{40}. Then add the square of \frac{33}{40} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+\frac{33}{20}v+\left(\frac{33}{40}\right)^{2}=\frac{9}{5}+\left(\frac{33}{40}\right)^{2}$$

Square \frac{33}{40} by squaring both the numerator and the denominator of the fraction.

$$v^{2}+\frac{33}{20}v+\frac{1089}{1600}=\frac{9}{5}+\frac{1089}{1600}$$

Add \frac{9}{5} to \frac{1089}{1600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$v^{2}+\frac{33}{20}v+\frac{1089}{1600}=\frac{3969}{1600}$$

Factor v^{2}+\frac{33}{20}v+\frac{1089}{1600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v+\frac{33}{40}\right)^{2}=\frac{3969}{1600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+\frac{33}{40}\right)^{2}}=\sqrt{\frac{3969}{1600}}$$

Simplify.

$$v+\frac{33}{40}=\frac{63}{40} v+\frac{33}{40}=-\frac{63}{40}$$

Subtract \frac{33}{40} from both sides of the equation.

$$v=\frac{3}{4} v=-\frac{12}{5}$$