Solve (x+3)^2+(7-x-2)^2=4

$$( x + 3 ) ^ { 2 } + ( 7 - x - 2 ) ^ { 2 } = 4$$

Solve for x (complex solution)

$$x=1+\sqrt{14}i\approx 1+3.741657387i$$

$$x=-\sqrt{14}i+1\approx 1-3.741657387i$$

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}.

$$x^{2}+6x+9+\left(7-x-2\right)^{2}=4$$

Subtract 2 from 7 to get 5.

$$x^{2}+6x+9+\left(5-x\right)^{2}=4$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-x\right)^{2}.

$$x^{2}+6x+9+25-10x+x^{2}=4$$

Add 9 and 25 to get 34.

$$x^{2}+6x+34-10x+x^{2}=4$$

Combine 6x and -10x to get -4x.

$$x^{2}-4x+34+x^{2}=4$$

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

$$2x^{2}-4x+34=4$$

Subtract 4 from both sides.

$$2x^{2}-4x+34-4=0$$

Subtract 4 from 34 to get 30.

$$2x^{2}-4x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\times 30}}{2\times 2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\times 30}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\times 30}}{2\times 2}$$

Multiply -8 times 30.

$$x=\frac{-\left(-4\right)±\sqrt{16-240}}{2\times 2}$$

Add 16 to -240.

$$x=\frac{-\left(-4\right)±\sqrt{-224}}{2\times 2}$$

Take the square root of -224.

$$x=\frac{-\left(-4\right)±4\sqrt{14}i}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{14}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±4\sqrt{14}i}{4}$$

Now solve the equation x=\frac{4±4\sqrt{14}i}{4} when ± is plus. Add 4 to 4i\sqrt{14}.

$$x=\frac{4+4\sqrt{14}i}{4}$$

Divide 4+4i\sqrt{14} by 4.

$$x=1+\sqrt{14}i$$

Now solve the equation x=\frac{4±4\sqrt{14}i}{4} when ± is minus. Subtract 4i\sqrt{14} from 4.

$$x=\frac{-4\sqrt{14}i+4}{4}$$

Divide 4-4i\sqrt{14} by 4.

$$x=-\sqrt{14}i+1$$

The equation is now solved.

$$x=1+\sqrt{14}i x=-\sqrt{14}i+1$$

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}.

$$x^{2}+6x+9+\left(7-x-2\right)^{2}=4$$

Subtract 2 from 7 to get 5.

$$x^{2}+6x+9+\left(5-x\right)^{2}=4$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-x\right)^{2}.

$$x^{2}+6x+9+25-10x+x^{2}=4$$

Add 9 and 25 to get 34.

$$x^{2}+6x+34-10x+x^{2}=4$$

Combine 6x and -10x to get -4x.

$$x^{2}-4x+34+x^{2}=4$$

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

$$2x^{2}-4x+34=4$$

Subtract 34 from both sides.

$$2x^{2}-4x=4-34$$

Subtract 34 from 4 to get -30.

$$2x^{2}-4x=-30$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=-\frac{30}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=-\frac{30}{2}$$

Divide -4 by 2.

$$x^{2}-2x=-\frac{30}{2}$$

Divide -30 by 2.

$$x^{2}-2x=-15$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=-15+1$$

Add -15 to 1.

$$x^{2}-2x+1=-14$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=-14$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-14}$$

Simplify.

$$x-1=\sqrt{14}i x-1=-\sqrt{14}i$$

Add 1 to both sides of the equation.

$$x=1+\sqrt{14}i x=-\sqrt{14}i+1$$