Solve 4x^2+40x+16800=0

$$4 { x }^{ 2 } +40x+16800=0$$

Solve for x (complex solution)

$$x=-5+5\sqrt{167}i\approx -5+64.614239917i$$

$$x=-5\sqrt{167}i-5\approx -5-64.614239917i$$

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.

$$4x^{2}+40x+16800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 40 for b, and 16800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40±\sqrt{40^{2}-4\times 4\times 16800}}{2\times 4}$$

Square 40.

$$x=\frac{-40±\sqrt{1600-4\times 4\times 16800}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-40±\sqrt{1600-16\times 16800}}{2\times 4}$$

Multiply -16 times 16800.

$$x=\frac{-40±\sqrt{1600-268800}}{2\times 4}$$

Add 1600 to -268800.

$$x=\frac{-40±\sqrt{-267200}}{2\times 4}$$

Take the square root of -267200.

$$x=\frac{-40±40\sqrt{167}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-40±40\sqrt{167}i}{8}$$

Now solve the equation x=\frac{-40±40\sqrt{167}i}{8} when ± is plus. Add -40 to 40i\sqrt{167}.

$$x=\frac{-40+40\sqrt{167}i}{8}$$

Divide -40+40i\sqrt{167} by 8.

$$x=-5+5\sqrt{167}i$$

Now solve the equation x=\frac{-40±40\sqrt{167}i}{8} when ± is minus. Subtract 40i\sqrt{167} from -40.

$$x=\frac{-40\sqrt{167}i-40}{8}$$

Divide -40-40i\sqrt{167} by 8.

$$x=-5\sqrt{167}i-5$$

The equation is now solved.

$$x=-5+5\sqrt{167}i x=-5\sqrt{167}i-5$$

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.

$$4x^{2}+40x+16800=0$$

Subtract 16800 from both sides of the equation.

$$4x^{2}+40x+16800-16800=-16800$$

Subtracting 16800 from itself leaves 0.

$$4x^{2}+40x=-16800$$

Divide both sides by 4.

$$\frac{4x^{2}+40x}{4}=-\frac{16800}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{40}{4}x=-\frac{16800}{4}$$

Divide 40 by 4.

$$x^{2}+10x=-\frac{16800}{4}$$

Divide -16800 by 4.

$$x^{2}+10x=-4200$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=-4200+5^{2}$$

Square 5.

$$x^{2}+10x+25=-4200+25$$

Add -4200 to 25.

$$x^{2}+10x+25=-4175$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=-4175$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{-4175}$$

Simplify.

$$x+5=5\sqrt{167}i x+5=-5\sqrt{167}i$$

Subtract 5 from both sides of the equation.

$$x=-5+5\sqrt{167}i x=-5\sqrt{167}i-5$$