Solve 1+36x/(x+3)^2=0

$$1 + \frac { 36 x } { ( x + 3 ) ^ { 2 } } = 0$$

Solve for x

$$x=12\sqrt{3}-21\approx -0.215390309$$

$$x=-12\sqrt{3}-21\approx -41.784609691$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by \left(x+3\right)^{2}.

$$\left(x+3\right)^{2}+36x=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+6x+9+36x=0$$

Combine 6x and 36x to get 42x.

$$x^{2}+42x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 42 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-42±\sqrt{42^{2}-4\times 9}}{2}$$

Square 42.

$$x=\frac{-42±\sqrt{1764-4\times 9}}{2}$$

Multiply -4 times 9.

$$x=\frac{-42±\sqrt{1764-36}}{2}$$

Add 1764 to -36.

$$x=\frac{-42±\sqrt{1728}}{2}$$

Take the square root of 1728.

$$x=\frac{-42±24\sqrt{3}}{2}$$

Now solve the equation x=\frac{-42±24\sqrt{3}}{2} when ± is plus. Add -42 to 24\sqrt{3}.

$$x=\frac{24\sqrt{3}-42}{2}$$

Divide -42+24\sqrt{3} by 2.

$$x=12\sqrt{3}-21$$

Now solve the equation x=\frac{-42±24\sqrt{3}}{2} when ± is minus. Subtract 24\sqrt{3} from -42.

$$x=\frac{-24\sqrt{3}-42}{2}$$

Divide -42-24\sqrt{3} by 2.

$$x=-12\sqrt{3}-21$$

The equation is now solved.

$$x=12\sqrt{3}-21 x=-12\sqrt{3}-21$$

Steps for Completing the Square

Variable x cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by \left(x+3\right)^{2}.

$$\left(x+3\right)^{2}+36x=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+6x+9+36x=0$$

Combine 6x and 36x to get 42x.

$$x^{2}+42x+9=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+42x=-9$$

Divide 42, the coefficient of the x term, by 2 to get 21. Then add the square of 21 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+42x+21^{2}=-9+21^{2}$$

Square 21.

$$x^{2}+42x+441=-9+441$$

Add -9 to 441.

$$x^{2}+42x+441=432$$

Factor x^{2}+42x+441. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+21\right)^{2}=432$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+21\right)^{2}}=\sqrt{432}$$

Simplify.

$$x+21=12\sqrt{3} x+21=-12\sqrt{3}$$

Subtract 21 from both sides of the equation.

$$x=12\sqrt{3}-21 x=-12\sqrt{3}-21$$