Solve 100x-10x^2=4000

$$100x-10 { x }^{ 2 } =4000$$

Solve for x (complex solution)

$$x=5+5\sqrt{15}i\approx 5+19.364916731i$$

$$x=-5\sqrt{15}i+5\approx 5-19.364916731i$$

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.

$$-10x^{2}+100x=4000$$

Subtract 4000 from both sides of the equation.

$$-10x^{2}+100x-4000=4000-4000$$

Subtracting 4000 from itself leaves 0.

$$-10x^{2}+100x-4000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 100 for b, and -4000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-10\right)\left(-4000\right)}}{2\left(-10\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-10\right)\left(-4000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-100±\sqrt{10000+40\left(-4000\right)}}{2\left(-10\right)}$$

Multiply 40 times -4000.

$$x=\frac{-100±\sqrt{10000-160000}}{2\left(-10\right)}$$

Add 10000 to -160000.

$$x=\frac{-100±\sqrt{-150000}}{2\left(-10\right)}$$

Take the square root of -150000.

$$x=\frac{-100±100\sqrt{15}i}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-100±100\sqrt{15}i}{-20}$$

Now solve the equation x=\frac{-100±100\sqrt{15}i}{-20} when ± is plus. Add -100 to 100i\sqrt{15}.

$$x=\frac{-100+100\sqrt{15}i}{-20}$$

Divide -100+100i\sqrt{15} by -20.

$$x=-5\sqrt{15}i+5$$

Now solve the equation x=\frac{-100±100\sqrt{15}i}{-20} when ± is minus. Subtract 100i\sqrt{15} from -100.

$$x=\frac{-100\sqrt{15}i-100}{-20}$$

Divide -100-100i\sqrt{15} by -20.

$$x=5+5\sqrt{15}i$$

The equation is now solved.

$$x=-5\sqrt{15}i+5 x=5+5\sqrt{15}i$$

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.

$$-10x^{2}+100x=4000$$

Divide both sides by -10.

$$\frac{-10x^{2}+100x}{-10}=\frac{4000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{100}{-10}x=\frac{4000}{-10}$$

Divide 100 by -10.

$$x^{2}-10x=\frac{4000}{-10}$$

Divide 4000 by -10.

$$x^{2}-10x=-400$$

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.

$$x^{2}-10x+\left(-5\right)^{2}=-400+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-400+25$$

Add -400 to 25.

$$x^{2}-10x+25=-375$$

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}.

$$\left(x-5\right)^{2}=-375$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{-375}$$

Simplify.

$$x-5=5\sqrt{15}i x-5=-5\sqrt{15}i$$

Add 5 to both sides of the equation.

$$x=5+5\sqrt{15}i x=-5\sqrt{15}i+5$$