Solve x^2+(3x+6)^2=4x^2

$$x ^ { 2 } + ( 3 x + 6 ) ^ { 2 } = 4 x ^ { 2 }$$

Solve for x

$$x=\sqrt{3}-3\approx -1.267949192$$

$$x=-\sqrt{3}-3\approx -4.732050808$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+6\right)^{2}.

$$x^{2}+9x^{2}+36x+36=4x^{2}$$

Combine x^{2} and 9x^{2} to get 10x^{2}.

$$10x^{2}+36x+36=4x^{2}$$

Subtract 4x^{2} from both sides.

$$10x^{2}+36x+36-4x^{2}=0$$

Combine 10x^{2} and -4x^{2} to get 6x^{2}.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 36 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-36±\sqrt{36^{2}-4\times 6\times 36}}{2\times 6}$$

Square 36.

$$x=\frac{-36±\sqrt{1296-4\times 6\times 36}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-36±\sqrt{1296-24\times 36}}{2\times 6}$$

Multiply -24 times 36.

$$x=\frac{-36±\sqrt{1296-864}}{2\times 6}$$

Add 1296 to -864.

$$x=\frac{-36±\sqrt{432}}{2\times 6}$$

Take the square root of 432.

$$x=\frac{-36±12\sqrt{3}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-36±12\sqrt{3}}{12}$$

Now solve the equation x=\frac{-36±12\sqrt{3}}{12} when ± is plus. Add -36 to 12\sqrt{3}.

$$x=\frac{12\sqrt{3}-36}{12}$$

Divide -36+12\sqrt{3} by 12.

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

Now solve the equation x=\frac{-36±12\sqrt{3}}{12} when ± is minus. Subtract 12\sqrt{3} from -36.

$$x=\frac{-12\sqrt{3}-36}{12}$$

Divide -36-12\sqrt{3} by 12.

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

The equation is now solved.

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

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+6\right)^{2}.

$$x^{2}+9x^{2}+36x+36=4x^{2}$$

Combine x^{2} and 9x^{2} to get 10x^{2}.

$$10x^{2}+36x+36=4x^{2}$$

Subtract 4x^{2} from both sides.

$$10x^{2}+36x+36-4x^{2}=0$$

Combine 10x^{2} and -4x^{2} to get 6x^{2}.

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

Subtract 36 from both sides. Anything subtracted from zero gives its negation.

$$6x^{2}+36x=-36$$

Divide both sides by 6.

$$\frac{6x^{2}+36x}{6}=-\frac{36}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{36}{6}x=-\frac{36}{6}$$

Divide 36 by 6.

$$x^{2}+6x=-\frac{36}{6}$$

Divide -36 by 6.

$$x^{2}+6x=-6$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=-6+3^{2}$$

Square 3.

$$x^{2}+6x+9=-6+9$$

Add -6 to 9.

$$x^{2}+6x+9=3$$

Factor x^{2}+6x+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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract 3 from both sides of the equation.

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