Solve -5x^2-360x-1980=6300

$$- 5 x ^ { 2 } - 360 x - 1980 = 6300$$

Solve for x (complex solution)

$$x=-36+6\sqrt{10}i\approx -36+18.973665961i$$

$$x=-6\sqrt{10}i-36\approx -36-18.973665961i$$

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.

$$-5x^{2}-360x-1980=6300$$

Subtract 6300 from both sides of the equation.

$$-5x^{2}-360x-1980-6300=6300-6300$$

Subtracting 6300 from itself leaves 0.

$$-5x^{2}-360x-1980-6300=0$$

Subtract 6300 from -1980.

$$-5x^{2}-360x-8280=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -360 for b, and -8280 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-360\right)±\sqrt{\left(-360\right)^{2}-4\left(-5\right)\left(-8280\right)}}{2\left(-5\right)}$$

Square -360.

$$x=\frac{-\left(-360\right)±\sqrt{129600-4\left(-5\right)\left(-8280\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-\left(-360\right)±\sqrt{129600+20\left(-8280\right)}}{2\left(-5\right)}$$

Multiply 20 times -8280.

$$x=\frac{-\left(-360\right)±\sqrt{129600-165600}}{2\left(-5\right)}$$

Add 129600 to -165600.

$$x=\frac{-\left(-360\right)±\sqrt{-36000}}{2\left(-5\right)}$$

Take the square root of -36000.

$$x=\frac{-\left(-360\right)±60\sqrt{10}i}{2\left(-5\right)}$$

The opposite of -360 is 360.

$$x=\frac{360±60\sqrt{10}i}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{360±60\sqrt{10}i}{-10}$$

Now solve the equation x=\frac{360±60\sqrt{10}i}{-10} when ± is plus. Add 360 to 60i\sqrt{10}.

$$x=\frac{360+60\sqrt{10}i}{-10}$$

Divide 360+60i\sqrt{10} by -10.

$$x=-6\sqrt{10}i-36$$

Now solve the equation x=\frac{360±60\sqrt{10}i}{-10} when ± is minus. Subtract 60i\sqrt{10} from 360.

$$x=\frac{-60\sqrt{10}i+360}{-10}$$

Divide 360-60i\sqrt{10} by -10.

$$x=-36+6\sqrt{10}i$$

The equation is now solved.

$$x=-6\sqrt{10}i-36 x=-36+6\sqrt{10}i$$

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.

$$-5x^{2}-360x-1980=6300$$

Add 1980 to both sides of the equation.

$$-5x^{2}-360x-1980-\left(-1980\right)=6300-\left(-1980\right)$$

Subtracting -1980 from itself leaves 0.

$$-5x^{2}-360x=6300-\left(-1980\right)$$

Subtract -1980 from 6300.

$$-5x^{2}-360x=8280$$

Divide both sides by -5.

$$\frac{-5x^{2}-360x}{-5}=\frac{8280}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\left(-\frac{360}{-5}\right)x=\frac{8280}{-5}$$

Divide -360 by -5.

$$x^{2}+72x=\frac{8280}{-5}$$

Divide 8280 by -5.

$$x^{2}+72x=-1656$$

Divide 72, the coefficient of the x term, by 2 to get 36. Then add the square of 36 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+72x+36^{2}=-1656+36^{2}$$

Square 36.

$$x^{2}+72x+1296=-1656+1296$$

Add -1656 to 1296.

$$x^{2}+72x+1296=-360$$

Factor x^{2}+72x+1296. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+36\right)^{2}=-360$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+36\right)^{2}}=\sqrt{-360}$$

Simplify.

$$x+36=6\sqrt{10}i x+36=-6\sqrt{10}i$$

Subtract 36 from both sides of the equation.

$$x=-36+6\sqrt{10}i x=-6\sqrt{10}i-36$$