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

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

Solve for x (complex solution)

$$x=\sqrt{7}-4\approx -1.354248689$$

$$x=-\left(\sqrt{7}+4\right)\approx -6.645751311$$

Steps Using the Quadratic Formula

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

$$2\left(x^{2}+6x+9\right)=-4x$$

Use the distributive property to multiply 2 by x^{2}+6x+9.

$$2x^{2}+12x+18=-4x$$

Add 4x to both sides.

$$2x^{2}+12x+18+4x=0$$

Combine 12x and 4x to get 16x.

$$2x^{2}+16x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 16 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 2\times 18}}{2\times 2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 2\times 18}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-16±\sqrt{256-8\times 18}}{2\times 2}$$

Multiply -8 times 18.

$$x=\frac{-16±\sqrt{256-144}}{2\times 2}$$

Add 256 to -144.

$$x=\frac{-16±\sqrt{112}}{2\times 2}$$

Take the square root of 112.

$$x=\frac{-16±4\sqrt{7}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-16±4\sqrt{7}}{4}$$

Now solve the equation x=\frac{-16±4\sqrt{7}}{4} when ± is plus. Add -16 to 4\sqrt{7}.

$$x=\frac{4\sqrt{7}-16}{4}$$

Divide -16+4\sqrt{7} by 4.

$$x=\sqrt{7}-4$$

Now solve the equation x=\frac{-16±4\sqrt{7}}{4} when ± is minus. Subtract 4\sqrt{7} from -16.

$$x=\frac{-4\sqrt{7}-16}{4}$$

Divide -16-4\sqrt{7} by 4.

$$x=-\sqrt{7}-4$$

The equation is now solved.

$$x=\sqrt{7}-4 x=-\sqrt{7}-4$$

Steps for Completing the Square

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

$$2\left(x^{2}+6x+9\right)=-4x$$

Use the distributive property to multiply 2 by x^{2}+6x+9.

$$2x^{2}+12x+18=-4x$$

Add 4x to both sides.

$$2x^{2}+12x+18+4x=0$$

Combine 12x and 4x to get 16x.

$$2x^{2}+16x+18=0$$

Subtract 18 from both sides. Anything subtracted from zero gives its negation.

$$2x^{2}+16x=-18$$

Divide both sides by 2.

$$\frac{2x^{2}+16x}{2}=-\frac{18}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{16}{2}x=-\frac{18}{2}$$

Divide 16 by 2.

$$x^{2}+8x=-\frac{18}{2}$$

Divide -18 by 2.

$$x^{2}+8x=-9$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=-9+4^{2}$$

Square 4.

$$x^{2}+8x+16=-9+16$$

Add -9 to 16.

$$x^{2}+8x+16=7$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=7$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{7}$$

Simplify.

$$x+4=\sqrt{7} x+4=-\sqrt{7}$$

Subtract 4 from both sides of the equation.

$$x=\sqrt{7}-4 x=-\sqrt{7}-4$$

Solve for x

$$x=\sqrt{7}-4\approx -1.354248689$$

$$x=-\sqrt{7}-4\approx -6.645751311$$

Steps Using the Quadratic Formula

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

$$2\left(x^{2}+6x+9\right)=-4x$$

Use the distributive property to multiply 2 by x^{2}+6x+9.

$$2x^{2}+12x+18=-4x$$

Add 4x to both sides.

$$2x^{2}+12x+18+4x=0$$

Combine 12x and 4x to get 16x.

$$2x^{2}+16x+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 16 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 2\times 18}}{2\times 2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 2\times 18}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-16±\sqrt{256-8\times 18}}{2\times 2}$$

Multiply -8 times 18.

$$x=\frac{-16±\sqrt{256-144}}{2\times 2}$$

Add 256 to -144.

$$x=\frac{-16±\sqrt{112}}{2\times 2}$$

Take the square root of 112.

$$x=\frac{-16±4\sqrt{7}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-16±4\sqrt{7}}{4}$$

Now solve the equation x=\frac{-16±4\sqrt{7}}{4} when ± is plus. Add -16 to 4\sqrt{7}.

$$x=\frac{4\sqrt{7}-16}{4}$$

Divide -16+4\sqrt{7} by 4.

$$x=\sqrt{7}-4$$

Now solve the equation x=\frac{-16±4\sqrt{7}}{4} when ± is minus. Subtract 4\sqrt{7} from -16.

$$x=\frac{-4\sqrt{7}-16}{4}$$

Divide -16-4\sqrt{7} by 4.

$$x=-\sqrt{7}-4$$

The equation is now solved.

$$x=\sqrt{7}-4 x=-\sqrt{7}-4$$

Steps for Completing the Square

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

$$2\left(x^{2}+6x+9\right)=-4x$$

Use the distributive property to multiply 2 by x^{2}+6x+9.

$$2x^{2}+12x+18=-4x$$

Add 4x to both sides.

$$2x^{2}+12x+18+4x=0$$

Combine 12x and 4x to get 16x.

$$2x^{2}+16x+18=0$$

Subtract 18 from both sides. Anything subtracted from zero gives its negation.

$$2x^{2}+16x=-18$$

Divide both sides by 2.

$$\frac{2x^{2}+16x}{2}=-\frac{18}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{16}{2}x=-\frac{18}{2}$$

Divide 16 by 2.

$$x^{2}+8x=-\frac{18}{2}$$

Divide -18 by 2.

$$x^{2}+8x=-9$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=-9+4^{2}$$

Square 4.

$$x^{2}+8x+16=-9+16$$

Add -9 to 16.

$$x^{2}+8x+16=7$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=7$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{7}$$

Simplify.

$$x+4=\sqrt{7} x+4=-\sqrt{7}$$

Subtract 4 from both sides of the equation.

$$x=\sqrt{7}-4 x=-\sqrt{7}-4$$