Solve (x+43)^2+(2x+34-8)^2=0

$$( x + 43 ) ^ { 2 } + ( 2 x + 34 - 8 ) ^ { 2 } = 0$$

Solve for x (complex solution)

$$x=-19+12i$$

$$x=-19-12i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+43\right)^{2}.

$$x^{2}+86x+1849+\left(2x+34-8\right)^{2}=0$$

Subtract 8 from 34 to get 26.

$$x^{2}+86x+1849+\left(2x+26\right)^{2}=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+26\right)^{2}.

$$x^{2}+86x+1849+4x^{2}+104x+676=0$$

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$5x^{2}+86x+1849+104x+676=0$$

Combine 86x and 104x to get 190x.

$$5x^{2}+190x+1849+676=0$$

Add 1849 and 676 to get 2525.

$$5x^{2}+190x+2525=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 190 for b, and 2525 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-190±\sqrt{190^{2}-4\times 5\times 2525}}{2\times 5}$$

Square 190.

$$x=\frac{-190±\sqrt{36100-4\times 5\times 2525}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-190±\sqrt{36100-20\times 2525}}{2\times 5}$$

Multiply -20 times 2525.

$$x=\frac{-190±\sqrt{36100-50500}}{2\times 5}$$

Add 36100 to -50500.

$$x=\frac{-190±\sqrt{-14400}}{2\times 5}$$

Take the square root of -14400.

$$x=\frac{-190±120i}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-190±120i}{10}$$

Now solve the equation x=\frac{-190±120i}{10} when ± is plus. Add -190 to 120i.

$$x=\frac{-190+120i}{10}$$

Divide -190+120i by 10.

$$x=-19+12i$$

Now solve the equation x=\frac{-190±120i}{10} when ± is minus. Subtract 120i from -190.

$$x=\frac{-190-120i}{10}$$

Divide -190-120i by 10.

$$x=-19-12i$$

The equation is now solved.

$$x=-19+12i x=-19-12i$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+43\right)^{2}.

$$x^{2}+86x+1849+\left(2x+34-8\right)^{2}=0$$

Subtract 8 from 34 to get 26.

$$x^{2}+86x+1849+\left(2x+26\right)^{2}=0$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+26\right)^{2}.

$$x^{2}+86x+1849+4x^{2}+104x+676=0$$

Combine x^{2} and 4x^{2} to get 5x^{2}.

$$5x^{2}+86x+1849+104x+676=0$$

Combine 86x and 104x to get 190x.

$$5x^{2}+190x+1849+676=0$$

Add 1849 and 676 to get 2525.

$$5x^{2}+190x+2525=0$$

Subtract 2525 from both sides. Anything subtracted from zero gives its negation.

$$5x^{2}+190x=-2525$$

Divide both sides by 5.

$$\frac{5x^{2}+190x}{5}=-\frac{2525}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{190}{5}x=-\frac{2525}{5}$$

Divide 190 by 5.

$$x^{2}+38x=-\frac{2525}{5}$$

Divide -2525 by 5.

$$x^{2}+38x=-505$$

Divide 38, the coefficient of the x term, by 2 to get 19. Then add the square of 19 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+38x+19^{2}=-505+19^{2}$$

Square 19.

$$x^{2}+38x+361=-505+361$$

Add -505 to 361.

$$x^{2}+38x+361=-144$$

Factor x^{2}+38x+361. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+19\right)^{2}=-144$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+19\right)^{2}}=\sqrt{-144}$$

Simplify.

$$x+19=12i x+19=-12i$$

Subtract 19 from both sides of the equation.

$$x=-19+12i x=-19-12i$$