Solve 9(x-2)^2=121(x+1)^2

$$9 ( x - 2 ) ^ { 2 } = 121 ( x + 1 ) ^ { 2 }$$

Solve for x

$$x=-\frac{5}{14}\approx -0.357142857$$

$$x = -\frac{17}{8} = -2\frac{1}{8} = -2.125$$

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}.

$$9\left(x^{2}-4x+4\right)=121\left(x+1\right)^{2}$$

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

$$9x^{2}-36x+36=121\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$9x^{2}-36x+36=121\left(x^{2}+2x+1\right)$$

Use the distributive property to multiply 121 by x^{2}+2x+1.

$$9x^{2}-36x+36=121x^{2}+242x+121$$

Subtract 121x^{2} from both sides.

$$9x^{2}-36x+36-121x^{2}=242x+121$$

Combine 9x^{2} and -121x^{2} to get -112x^{2}.

$$-112x^{2}-36x+36=242x+121$$

Subtract 242x from both sides.

$$-112x^{2}-36x+36-242x=121$$

Combine -36x and -242x to get -278x.

$$-112x^{2}-278x+36=121$$

Subtract 121 from both sides.

$$-112x^{2}-278x+36-121=0$$

Subtract 121 from 36 to get -85.

$$-112x^{2}-278x-85=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -112 for a, -278 for b, and -85 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-278\right)±\sqrt{\left(-278\right)^{2}-4\left(-112\right)\left(-85\right)}}{2\left(-112\right)}$$

Square -278.

$$x=\frac{-\left(-278\right)±\sqrt{77284-4\left(-112\right)\left(-85\right)}}{2\left(-112\right)}$$

Multiply -4 times -112.

$$x=\frac{-\left(-278\right)±\sqrt{77284+448\left(-85\right)}}{2\left(-112\right)}$$

Multiply 448 times -85.

$$x=\frac{-\left(-278\right)±\sqrt{77284-38080}}{2\left(-112\right)}$$

Add 77284 to -38080.

$$x=\frac{-\left(-278\right)±\sqrt{39204}}{2\left(-112\right)}$$

Take the square root of 39204.

$$x=\frac{-\left(-278\right)±198}{2\left(-112\right)}$$

The opposite of -278 is 278.

$$x=\frac{278±198}{2\left(-112\right)}$$

Multiply 2 times -112.

$$x=\frac{278±198}{-224}$$

Now solve the equation x=\frac{278±198}{-224} when ± is plus. Add 278 to 198.

$$x=\frac{476}{-224}$$

Reduce the fraction \frac{476}{-224} to lowest terms by extracting and canceling out 28.

$$x=-\frac{17}{8}$$

Now solve the equation x=\frac{278±198}{-224} when ± is minus. Subtract 198 from 278.

$$x=\frac{80}{-224}$$

Reduce the fraction \frac{80}{-224} to lowest terms by extracting and canceling out 16.

$$x=-\frac{5}{14}$$

The equation is now solved.

$$x=-\frac{17}{8} x=-\frac{5}{14}$$

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}.

$$9\left(x^{2}-4x+4\right)=121\left(x+1\right)^{2}$$

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

$$9x^{2}-36x+36=121\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$9x^{2}-36x+36=121\left(x^{2}+2x+1\right)$$

Use the distributive property to multiply 121 by x^{2}+2x+1.

$$9x^{2}-36x+36=121x^{2}+242x+121$$

Subtract 121x^{2} from both sides.

$$9x^{2}-36x+36-121x^{2}=242x+121$$

Combine 9x^{2} and -121x^{2} to get -112x^{2}.

$$-112x^{2}-36x+36=242x+121$$

Subtract 242x from both sides.

$$-112x^{2}-36x+36-242x=121$$

Combine -36x and -242x to get -278x.

$$-112x^{2}-278x+36=121$$

Subtract 36 from both sides.

$$-112x^{2}-278x=121-36$$

Subtract 36 from 121 to get 85.

$$-112x^{2}-278x=85$$

Divide both sides by -112.

$$\frac{-112x^{2}-278x}{-112}=\frac{85}{-112}$$

Dividing by -112 undoes the multiplication by -112.

$$x^{2}+\left(-\frac{278}{-112}\right)x=\frac{85}{-112}$$

Reduce the fraction \frac{-278}{-112} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{139}{56}x=\frac{85}{-112}$$

Divide 85 by -112.

$$x^{2}+\frac{139}{56}x=-\frac{85}{112}$$

Divide \frac{139}{56}, the coefficient of the x term, by 2 to get \frac{139}{112}. Then add the square of \frac{139}{112} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{139}{56}x+\left(\frac{139}{112}\right)^{2}=-\frac{85}{112}+\left(\frac{139}{112}\right)^{2}$$

Square \frac{139}{112} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{139}{56}x+\frac{19321}{12544}=-\frac{85}{112}+\frac{19321}{12544}$$

Add -\frac{85}{112} to \frac{19321}{12544} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{139}{56}x+\frac{19321}{12544}=\frac{9801}{12544}$$

Factor x^{2}+\frac{139}{56}x+\frac{19321}{12544}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{139}{112}\right)^{2}=\frac{9801}{12544}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{139}{112}\right)^{2}}=\sqrt{\frac{9801}{12544}}$$

Simplify.

$$x+\frac{139}{112}=\frac{99}{112} x+\frac{139}{112}=-\frac{99}{112}$$

Subtract \frac{139}{112} from both sides of the equation.

$$x=-\frac{5}{14} x=-\frac{17}{8}$$