Solve 9(x-2)^2-121=0

$$9 ( x - 2 ) ^ { 2 } - 121 = 0$$

Solve for x

$$x = \frac{17}{3} = 5\frac{2}{3} \approx 5.666666667$$

$$x = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$9\left(x^{2}-4x+4\right)-121=0$$

Use the distributive property to multiply 9 by x^{2}-4x+4.

$$9x^{2}-36x+36-121=0$$

Subtract 121 from 36 to get -85.

$$9x^{2}-36x-85=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9x^{2}+ax+bx-85. To find a and b, set up a system to be solved.

$$a+b=-36 ab=9\left(-85\right)=-765$$

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 -765.

$$1,-765 3,-255 5,-153 9,-85 15,-51 17,-45$$

Calculate the sum for each pair.

$$1-765=-764 3-255=-252 5-153=-148 9-85=-76 15-51=-36 17-45=-28$$

The solution is the pair that gives sum -36.

$$a=-51 b=15$$

Rewrite 9x^{2}-36x-85 as \left(9x^{2}-51x\right)+\left(15x-85\right).

$$\left(9x^{2}-51x\right)+\left(15x-85\right)$$

Factor out 3x in the first and 5 in the second group.

$$3x\left(3x-17\right)+5\left(3x-17\right)$$

Factor out common term 3x-17 by using distributive property.

$$\left(3x-17\right)\left(3x+5\right)$$

To find equation solutions, solve 3x-17=0 and 3x+5=0.

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

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$9\left(x^{2}-4x+4\right)-121=0$$

Use the distributive property to multiply 9 by x^{2}-4x+4.

$$9x^{2}-36x+36-121=0$$

Subtract 121 from 36 to get -85.

$$9x^{2}-36x-85=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -36 for b, and -85 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 9\left(-85\right)}}{2\times 9}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 9\left(-85\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-36\right)±\sqrt{1296-36\left(-85\right)}}{2\times 9}$$

Multiply -36 times -85.

$$x=\frac{-\left(-36\right)±\sqrt{1296+3060}}{2\times 9}$$

Add 1296 to 3060.

$$x=\frac{-\left(-36\right)±\sqrt{4356}}{2\times 9}$$

Take the square root of 4356.

$$x=\frac{-\left(-36\right)±66}{2\times 9}$$

The opposite of -36 is 36.

$$x=\frac{36±66}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{36±66}{18}$$

Now solve the equation x=\frac{36±66}{18} when ± is plus. Add 36 to 66.

$$x=\frac{102}{18}$$

Reduce the fraction \frac{102}{18} to lowest terms by extracting and canceling out 6.

$$x=\frac{17}{3}$$

Now solve the equation x=\frac{36±66}{18} when ± is minus. Subtract 66 from 36.

$$x=-\frac{30}{18}$$

Reduce the fraction \frac{-30}{18} to lowest terms by extracting and canceling out 6.

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

The equation is now solved.

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

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$9\left(x^{2}-4x+4\right)-121=0$$

Use the distributive property to multiply 9 by x^{2}-4x+4.

$$9x^{2}-36x+36-121=0$$

Subtract 121 from 36 to get -85.

$$9x^{2}-36x-85=0$$

Add 85 to both sides. Anything plus zero gives itself.

$$9x^{2}-36x=85$$

Divide both sides by 9.

$$\frac{9x^{2}-36x}{9}=\frac{85}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\left(-\frac{36}{9}\right)x=\frac{85}{9}$$

Divide -36 by 9.

$$x^{2}-4x=\frac{85}{9}$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=\frac{85}{9}+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=\frac{85}{9}+4$$

Add \frac{85}{9} to 4.

$$x^{2}-4x+4=\frac{121}{9}$$

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

$$\left(x-2\right)^{2}=\frac{121}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{121}{9}}$$

Simplify.

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

Add 2 to both sides of the equation.

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