Solve -left(x+3right)^2-4(3x+1)=0

$$- { \left(x+3 \right) }^{ 2 } -4(3x+1)=0$$

Solve for x

$$x=2\sqrt{17}-9\approx -0.753788749$$

$$x=-2\sqrt{17}-9\approx -17.246211251$$

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}.

$$-\left(x^{2}+6x+9\right)-4\left(3x+1\right)=0$$

To find the opposite of x^{2}+6x+9, find the opposite of each term.

$$-x^{2}-6x-9-4\left(3x+1\right)=0$$

Use the distributive property to multiply -4 by 3x+1.

$$-x^{2}-6x-9-12x-4=0$$

Combine -6x and -12x to get -18x.

$$-x^{2}-18x-9-4=0$$

Subtract 4 from -9 to get -13.

$$-x^{2}-18x-13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -18 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-1\right)\left(-13\right)}}{2\left(-1\right)}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\left(-1\right)\left(-13\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-18\right)±\sqrt{324+4\left(-13\right)}}{2\left(-1\right)}$$

Multiply 4 times -13.

$$x=\frac{-\left(-18\right)±\sqrt{324-52}}{2\left(-1\right)}$$

Add 324 to -52.

$$x=\frac{-\left(-18\right)±\sqrt{272}}{2\left(-1\right)}$$

Take the square root of 272.

$$x=\frac{-\left(-18\right)±4\sqrt{17}}{2\left(-1\right)}$$

The opposite of -18 is 18.

$$x=\frac{18±4\sqrt{17}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{18±4\sqrt{17}}{-2}$$

Now solve the equation x=\frac{18±4\sqrt{17}}{-2} when ± is plus. Add 18 to 4\sqrt{17}.

$$x=\frac{4\sqrt{17}+18}{-2}$$

Divide 18+4\sqrt{17} by -2.

$$x=-2\sqrt{17}-9$$

Now solve the equation x=\frac{18±4\sqrt{17}}{-2} when ± is minus. Subtract 4\sqrt{17} from 18.

$$x=\frac{18-4\sqrt{17}}{-2}$$

Divide 18-4\sqrt{17} by -2.

$$x=2\sqrt{17}-9$$

The equation is now solved.

$$x=-2\sqrt{17}-9 x=2\sqrt{17}-9$$

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}.

$$-\left(x^{2}+6x+9\right)-4\left(3x+1\right)=0$$

To find the opposite of x^{2}+6x+9, find the opposite of each term.

$$-x^{2}-6x-9-4\left(3x+1\right)=0$$

Use the distributive property to multiply -4 by 3x+1.

$$-x^{2}-6x-9-12x-4=0$$

Combine -6x and -12x to get -18x.

$$-x^{2}-18x-9-4=0$$

Subtract 4 from -9 to get -13.

$$-x^{2}-18x-13=0$$

Add 13 to both sides. Anything plus zero gives itself.

$$-x^{2}-18x=13$$

Divide both sides by -1.

$$\frac{-x^{2}-18x}{-1}=\frac{13}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{18}{-1}\right)x=\frac{13}{-1}$$

Divide -18 by -1.

$$x^{2}+18x=\frac{13}{-1}$$

Divide 13 by -1.

$$x^{2}+18x=-13$$

Divide 18, the coefficient of the x term, by 2 to get 9. Then add the square of 9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+18x+9^{2}=-13+9^{2}$$

Square 9.

$$x^{2}+18x+81=-13+81$$

Add -13 to 81.

$$x^{2}+18x+81=68$$

Factor x^{2}+18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+9\right)^{2}=68$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{68}$$

Simplify.

$$x+9=2\sqrt{17} x+9=-2\sqrt{17}$$

Subtract 9 from both sides of the equation.

$$x=2\sqrt{17}-9 x=-2\sqrt{17}-9$$