Solve x^2+40x+7200=0

$${ x }^{ 2 } +40x+7200=0$$

Solve for x (complex solution)

$$x=-20+20\sqrt{17}i\approx -20+82.462112512i$$

$$x=-20\sqrt{17}i-20\approx -20-82.462112512i$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+40x+7200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 40 for b, and 7200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40±\sqrt{40^{2}-4\times 7200}}{2}$$

Square 40.

$$x=\frac{-40±\sqrt{1600-4\times 7200}}{2}$$

Multiply -4 times 7200.

$$x=\frac{-40±\sqrt{1600-28800}}{2}$$

Add 1600 to -28800.

$$x=\frac{-40±\sqrt{-27200}}{2}$$

Take the square root of -27200.

$$x=\frac{-40±40\sqrt{17}i}{2}$$

Now solve the equation x=\frac{-40±40\sqrt{17}i}{2} when ± is plus. Add -40 to 40i\sqrt{17}.

$$x=\frac{-40+40\sqrt{17}i}{2}$$

Divide -40+40i\sqrt{17} by 2.

$$x=-20+20\sqrt{17}i$$

Now solve the equation x=\frac{-40±40\sqrt{17}i}{2} when ± is minus. Subtract 40i\sqrt{17} from -40.

$$x=\frac{-40\sqrt{17}i-40}{2}$$

Divide -40-40i\sqrt{17} by 2.

$$x=-20\sqrt{17}i-20$$

The equation is now solved.

$$x=-20+20\sqrt{17}i x=-20\sqrt{17}i-20$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+40x+7200=0$$

Subtract 7200 from both sides of the equation.

$$x^{2}+40x+7200-7200=-7200$$

Subtracting 7200 from itself leaves 0.

$$x^{2}+40x=-7200$$

Divide 40, the coefficient of the x term, by 2 to get 20. Then add the square of 20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+40x+20^{2}=-7200+20^{2}$$

Square 20.

$$x^{2}+40x+400=-7200+400$$

Add -7200 to 400.

$$x^{2}+40x+400=-6800$$

Factor x^{2}+40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+20\right)^{2}=-6800$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+20\right)^{2}}=\sqrt{-6800}$$

Simplify.

$$x+20=20\sqrt{17}i x+20=-20\sqrt{17}i$$

Subtract 20 from both sides of the equation.

$$x=-20+20\sqrt{17}i x=-20\sqrt{17}i-20$$