Solve x^2/9-4left(x+1right)^2=0

$$\frac{ { x }^{ 2 } }{ 9 } -4 { \left(x+1 \right) }^{ 2 } =0$$

Solve for x

$$x = -\frac{6}{5} = -1\frac{1}{5} = -1.2$$

$$x=-\frac{6}{7}\approx -0.857142857$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 9.

$$x^{2}-36\left(x+1\right)^{2}=0$$

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

$$x^{2}-36\left(x^{2}+2x+1\right)=0$$

Use the distributive property to multiply -36 by x^{2}+2x+1.

$$x^{2}-36x^{2}-72x-36=0$$

Combine x^{2} and -36x^{2} to get -35x^{2}.

$$-35x^{2}-72x-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -35 for a, -72 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\left(-35\right)\left(-36\right)}}{2\left(-35\right)}$$

Square -72.

$$x=\frac{-\left(-72\right)±\sqrt{5184-4\left(-35\right)\left(-36\right)}}{2\left(-35\right)}$$

Multiply -4 times -35.

$$x=\frac{-\left(-72\right)±\sqrt{5184+140\left(-36\right)}}{2\left(-35\right)}$$

Multiply 140 times -36.

$$x=\frac{-\left(-72\right)±\sqrt{5184-5040}}{2\left(-35\right)}$$

Add 5184 to -5040.

$$x=\frac{-\left(-72\right)±\sqrt{144}}{2\left(-35\right)}$$

Take the square root of 144.

$$x=\frac{-\left(-72\right)±12}{2\left(-35\right)}$$

The opposite of -72 is 72.

$$x=\frac{72±12}{2\left(-35\right)}$$

Multiply 2 times -35.

$$x=\frac{72±12}{-70}$$

Now solve the equation x=\frac{72±12}{-70} when ± is plus. Add 72 to 12.

$$x=\frac{84}{-70}$$

Reduce the fraction \frac{84}{-70} to lowest terms by extracting and canceling out 14.

$$x=-\frac{6}{5}$$

Now solve the equation x=\frac{72±12}{-70} when ± is minus. Subtract 12 from 72.

$$x=\frac{60}{-70}$$

Reduce the fraction \frac{60}{-70} to lowest terms by extracting and canceling out 10.

$$x=-\frac{6}{7}$$

The equation is now solved.

$$x=-\frac{6}{5} x=-\frac{6}{7}$$

Steps for Completing the Square

Multiply both sides of the equation by 9.

$$x^{2}-36\left(x+1\right)^{2}=0$$

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

$$x^{2}-36\left(x^{2}+2x+1\right)=0$$

Use the distributive property to multiply -36 by x^{2}+2x+1.

$$x^{2}-36x^{2}-72x-36=0$$

Combine x^{2} and -36x^{2} to get -35x^{2}.

$$-35x^{2}-72x-36=0$$

Add 36 to both sides. Anything plus zero gives itself.

$$-35x^{2}-72x=36$$

Divide both sides by -35.

$$\frac{-35x^{2}-72x}{-35}=\frac{36}{-35}$$

Dividing by -35 undoes the multiplication by -35.

$$x^{2}+\left(-\frac{72}{-35}\right)x=\frac{36}{-35}$$

Divide -72 by -35.

$$x^{2}+\frac{72}{35}x=\frac{36}{-35}$$

Divide 36 by -35.

$$x^{2}+\frac{72}{35}x=-\frac{36}{35}$$

Divide \frac{72}{35}, the coefficient of the x term, by 2 to get \frac{36}{35}. Then add the square of \frac{36}{35} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{72}{35}x+\left(\frac{36}{35}\right)^{2}=-\frac{36}{35}+\left(\frac{36}{35}\right)^{2}$$

Square \frac{36}{35} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{72}{35}x+\frac{1296}{1225}=-\frac{36}{35}+\frac{1296}{1225}$$

Add -\frac{36}{35} to \frac{1296}{1225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{72}{35}x+\frac{1296}{1225}=\frac{36}{1225}$$

Factor x^{2}+\frac{72}{35}x+\frac{1296}{1225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{36}{35}\right)^{2}=\frac{36}{1225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{36}{35}\right)^{2}}=\sqrt{\frac{36}{1225}}$$

Simplify.

$$x+\frac{36}{35}=\frac{6}{35} x+\frac{36}{35}=-\frac{6}{35}$$

Subtract \frac{36}{35} from both sides of the equation.

$$x=-\frac{6}{7} x=-\frac{6}{5}$$