Solve (6x-5)(3x+2)-(9x+6)(3x+9)=0

$$( 6 x - 5 ) ( 3 x + 2 ) - ( 9 x + 6 ) ( 3 x + 9 ) = 0$$

Solve for x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x = -\frac{32}{3} = -10\frac{2}{3} \approx -10.666666667$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 6x-5 by 3x+2 and combine like terms.

$$18x^{2}-3x-10-\left(9x+6\right)\left(3x+9\right)=0$$

Use the distributive property to multiply 9x+6 by 3x+9 and combine like terms.

$$18x^{2}-3x-10-\left(27x^{2}+99x+54\right)=0$$

To find the opposite of 27x^{2}+99x+54, find the opposite of each term.

$$18x^{2}-3x-10-27x^{2}-99x-54=0$$

Combine 18x^{2} and -27x^{2} to get -9x^{2}.

$$-9x^{2}-3x-10-99x-54=0$$

Combine -3x and -99x to get -102x.

$$-9x^{2}-102x-10-54=0$$

Subtract 54 from -10 to get -64.

$$-9x^{2}-102x-64=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-64. To find a and b, set up a system to be solved.

$$a+b=-102 ab=-9\left(-64\right)=576$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 576.

$$-1,-576 -2,-288 -3,-192 -4,-144 -6,-96 -8,-72 -9,-64 -12,-48 -16,-36 -18,-32 -24,-24$$

Calculate the sum for each pair.

$$-1-576=-577 -2-288=-290 -3-192=-195 -4-144=-148 -6-96=-102 -8-72=-80 -9-64=-73 -12-48=-60 -16-36=-52 -18-32=-50 -24-24=-48$$

The solution is the pair that gives sum -102.

$$a=-6 b=-96$$

Rewrite -9x^{2}-102x-64 as \left(-9x^{2}-6x\right)+\left(-96x-64\right).

$$\left(-9x^{2}-6x\right)+\left(-96x-64\right)$$

Factor out 3x in the first and 32 in the second group.

$$3x\left(-3x-2\right)+32\left(-3x-2\right)$$

Factor out common term -3x-2 by using distributive property.

$$\left(-3x-2\right)\left(3x+32\right)$$

To find equation solutions, solve -3x-2=0 and 3x+32=0.

$$x=-\frac{2}{3} x=-\frac{32}{3}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6x-5 by 3x+2 and combine like terms.

$$18x^{2}-3x-10-\left(9x+6\right)\left(3x+9\right)=0$$

Use the distributive property to multiply 9x+6 by 3x+9 and combine like terms.

$$18x^{2}-3x-10-\left(27x^{2}+99x+54\right)=0$$

To find the opposite of 27x^{2}+99x+54, find the opposite of each term.

$$18x^{2}-3x-10-27x^{2}-99x-54=0$$

Combine 18x^{2} and -27x^{2} to get -9x^{2}.

$$-9x^{2}-3x-10-99x-54=0$$

Combine -3x and -99x to get -102x.

$$-9x^{2}-102x-10-54=0$$

Subtract 54 from -10 to get -64.

$$-9x^{2}-102x-64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -102 for b, and -64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-102\right)±\sqrt{\left(-102\right)^{2}-4\left(-9\right)\left(-64\right)}}{2\left(-9\right)}$$

Square -102.

$$x=\frac{-\left(-102\right)±\sqrt{10404-4\left(-9\right)\left(-64\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-102\right)±\sqrt{10404+36\left(-64\right)}}{2\left(-9\right)}$$

Multiply 36 times -64.

$$x=\frac{-\left(-102\right)±\sqrt{10404-2304}}{2\left(-9\right)}$$

Add 10404 to -2304.

$$x=\frac{-\left(-102\right)±\sqrt{8100}}{2\left(-9\right)}$$

Take the square root of 8100.

$$x=\frac{-\left(-102\right)±90}{2\left(-9\right)}$$

The opposite of -102 is 102.

$$x=\frac{102±90}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{102±90}{-18}$$

Now solve the equation x=\frac{102±90}{-18} when ± is plus. Add 102 to 90.

$$x=\frac{192}{-18}$$

Reduce the fraction \frac{192}{-18} to lowest terms by extracting and canceling out 6.

$$x=-\frac{32}{3}$$

Now solve the equation x=\frac{102±90}{-18} when ± is minus. Subtract 90 from 102.

$$x=\frac{12}{-18}$$

Reduce the fraction \frac{12}{-18} to lowest terms by extracting and canceling out 6.

$$x=-\frac{2}{3}$$

The equation is now solved.

$$x=-\frac{32}{3} x=-\frac{2}{3}$$

Steps for Completing the Square

Use the distributive property to multiply 6x-5 by 3x+2 and combine like terms.

$$18x^{2}-3x-10-\left(9x+6\right)\left(3x+9\right)=0$$

Use the distributive property to multiply 9x+6 by 3x+9 and combine like terms.

$$18x^{2}-3x-10-\left(27x^{2}+99x+54\right)=0$$

To find the opposite of 27x^{2}+99x+54, find the opposite of each term.

$$18x^{2}-3x-10-27x^{2}-99x-54=0$$

Combine 18x^{2} and -27x^{2} to get -9x^{2}.

$$-9x^{2}-3x-10-99x-54=0$$

Combine -3x and -99x to get -102x.

$$-9x^{2}-102x-10-54=0$$

Subtract 54 from -10 to get -64.

$$-9x^{2}-102x-64=0$$

Add 64 to both sides. Anything plus zero gives itself.

$$-9x^{2}-102x=64$$

Divide both sides by -9.

$$\frac{-9x^{2}-102x}{-9}=\frac{64}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\left(-\frac{102}{-9}\right)x=\frac{64}{-9}$$

Reduce the fraction \frac{-102}{-9} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{34}{3}x=\frac{64}{-9}$$

Divide 64 by -9.

$$x^{2}+\frac{34}{3}x=-\frac{64}{9}$$

Divide \frac{34}{3}, the coefficient of the x term, by 2 to get \frac{17}{3}. Then add the square of \frac{17}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{34}{3}x+\left(\frac{17}{3}\right)^{2}=-\frac{64}{9}+\left(\frac{17}{3}\right)^{2}$$

Square \frac{17}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{34}{3}x+\frac{289}{9}=\frac{-64+289}{9}$$

Add -\frac{64}{9} to \frac{289}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{34}{3}x+\frac{289}{9}=25$$

Factor x^{2}+\frac{34}{3}x+\frac{289}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{17}{3}\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{17}{3}\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+\frac{17}{3}=5 x+\frac{17}{3}=-5$$

Subtract \frac{17}{3} from both sides of the equation.

$$x=-\frac{2}{3} x=-\frac{32}{3}$$