Solve 0=(3x-7)(3x+11)

$$0 = ( 3 x - 7 ) ( 3 x + 11 )$$

Solve for x

$$x = -\frac{11}{3} = -3\frac{2}{3} \approx -3.666666667$$

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2.333333333$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 3x-7 by 3x+11 and combine like terms.

$$0=9x^{2}+12x-77$$

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

$$9x^{2}+12x-77=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9x^{2}+ax+bx-77. To find a and b, set up a system to be solved.

$$a+b=12 ab=9\left(-77\right)=-693$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -693.

$$-1,693 -3,231 -7,99 -9,77 -11,63 -21,33$$

Calculate the sum for each pair.

$$-1+693=692 -3+231=228 -7+99=92 -9+77=68 -11+63=52 -21+33=12$$

The solution is the pair that gives sum 12.

$$a=-21 b=33$$

Rewrite 9x^{2}+12x-77 as \left(9x^{2}-21x\right)+\left(33x-77\right).

$$\left(9x^{2}-21x\right)+\left(33x-77\right)$$

Factor out 3x in the first and 11 in the second group.

$$3x\left(3x-7\right)+11\left(3x-7\right)$$

Factor out common term 3x-7 by using distributive property.

$$\left(3x-7\right)\left(3x+11\right)$$

To find equation solutions, solve 3x-7=0 and 3x+11=0.

$$x=\frac{7}{3} x=-\frac{11}{3}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x-7 by 3x+11 and combine like terms.

$$0=9x^{2}+12x-77$$

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

$$9x^{2}+12x-77=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 12 for b, and -77 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 9\left(-77\right)}}{2\times 9}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 9\left(-77\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-12±\sqrt{144-36\left(-77\right)}}{2\times 9}$$

Multiply -36 times -77.

$$x=\frac{-12±\sqrt{144+2772}}{2\times 9}$$

Add 144 to 2772.

$$x=\frac{-12±\sqrt{2916}}{2\times 9}$$

Take the square root of 2916.

$$x=\frac{-12±54}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-12±54}{18}$$

Now solve the equation x=\frac{-12±54}{18} when ± is plus. Add -12 to 54.

$$x=\frac{42}{18}$$

Reduce the fraction \frac{42}{18} to lowest terms by extracting and canceling out 6.

$$x=\frac{7}{3}$$

Now solve the equation x=\frac{-12±54}{18} when ± is minus. Subtract 54 from -12.

$$x=-\frac{66}{18}$$

Reduce the fraction \frac{-66}{18} to lowest terms by extracting and canceling out 6.

$$x=-\frac{11}{3}$$

The equation is now solved.

$$x=\frac{7}{3} x=-\frac{11}{3}$$

Steps for Completing the Square

Use the distributive property to multiply 3x-7 by 3x+11 and combine like terms.

$$0=9x^{2}+12x-77$$

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

$$9x^{2}+12x-77=0$$

Add 77 to both sides. Anything plus zero gives itself.

$$9x^{2}+12x=77$$

Divide both sides by 9.

$$\frac{9x^{2}+12x}{9}=\frac{77}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\frac{12}{9}x=\frac{77}{9}$$

Reduce the fraction \frac{12}{9} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{4}{3}x=\frac{77}{9}$$

Divide \frac{4}{3}, the coefficient of the x term, by 2 to get \frac{2}{3}. Then add the square of \frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{3}x+\left(\frac{2}{3}\right)^{2}=\frac{77}{9}+\left(\frac{2}{3}\right)^{2}$$

Square \frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{77+4}{9}$$

Add \frac{77}{9} to \frac{4}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=9$$

Factor x^{2}+\frac{4}{3}x+\frac{4}{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}.

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

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{3}\right)^{2}}=\sqrt{9}$$

Simplify.

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

Subtract \frac{2}{3} from both sides of the equation.

$$x=\frac{7}{3} x=-\frac{11}{3}$$