Solve {l}{x+y+6=0}{x^2+y^2=4}

$$\left. \begin{array} { l } { x + y + 6 = 0 } \\ { x ^ { 2 } + y ^ { 2 } = 4 } \end{array} \right.$$

Solve for x, y (complex solution)

$$x=-3+\sqrt{7}i\approx -3+2.645751311i\text{, }y=-\sqrt{7}i-3\approx -3-2.645751311i$$

$$x=-\sqrt{7}i-3\approx -3-2.645751311i\text{, }y=-3+\sqrt{7}i\approx -3+2.645751311i$$

Steps Using Substitution and Quadratic Formula

Solve x+y+6=0 for x by isolating x on the left hand side of the equal sign.

$$x+y+6=0$$

Subtract 6 from both sides of the equation.

$$x+y=-6$$

Subtract y from both sides of the equation.

$$x=-y-6$$

Substitute -y-6 for x in the other equation, y^{2}+x^{2}=4.

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

Square -y-6.

$$y^{2}+y^{2}+12y+36=4$$

Add y^{2} to y^{2}.

$$2y^{2}+12y+36=4$$

Subtract 4 from both sides of the equation.

$$2y^{2}+12y+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1+1\left(-1\right)^{2} for a, 1\left(-6\right)\left(-1\right)\times 2 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-12±\sqrt{12^{2}-4\times 2\times 32}}{2\times 2}$$

Square 1\left(-6\right)\left(-1\right)\times 2.

$$y=\frac{-12±\sqrt{144-4\times 2\times 32}}{2\times 2}$$

Multiply -4 times 1+1\left(-1\right)^{2}.

$$y=\frac{-12±\sqrt{144-8\times 32}}{2\times 2}$$

Multiply -8 times 32.

$$y=\frac{-12±\sqrt{144-256}}{2\times 2}$$

Add 144 to -256.

$$y=\frac{-12±\sqrt{-112}}{2\times 2}$$

Take the square root of -112.

$$y=\frac{-12±4\sqrt{7}i}{2\times 2}$$

Multiply 2 times 1+1\left(-1\right)^{2}.

$$y=\frac{-12±4\sqrt{7}i}{4}$$

Now solve the equation y=\frac{-12±4\sqrt{7}i}{4} when ± is plus. Add -12 to 4i\sqrt{7}.

$$y=\frac{-12+4\sqrt{7}i}{4}$$

Divide -12+4i\sqrt{7} by 4.

$$y=-3+\sqrt{7}i$$

Now solve the equation y=\frac{-12±4\sqrt{7}i}{4} when ± is minus. Subtract 4i\sqrt{7} from -12.

$$y=\frac{-4\sqrt{7}i-12}{4}$$

Divide -12-4i\sqrt{7} by 4.

$$y=-\sqrt{7}i-3$$

There are two solutions for y: -3+i\sqrt{7} and -3-i\sqrt{7}. Substitute -3+i\sqrt{7} for y in the equation x=-y-6 to find the corresponding solution for x that satisfies both equations.

$$x=-\left(-3+\sqrt{7}i\right)-6$$

Now substitute -3-i\sqrt{7} for y in the equation x=-y-6 and solve to find the corresponding solution for x that satisfies both equations.

$$x=-\left(-\sqrt{7}i-3\right)-6$$

The system is now solved.

$$x=-\left(-3+\sqrt{7}i\right)-6,y=-3+\sqrt{7}i\text{ or }x=-\left(-\sqrt{7}i-3\right)-6,y=-\sqrt{7}i-3$$