Solve 100x-2100-x^2=0

$$100 x - 2100 - x ^ { 2 } = 0$$

Solve for x

$$x=30$$

$$x=70$$

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.

$$-x^{2}+100x-2100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 100 for b, and -2100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-1\right)\left(-2100\right)}}{2\left(-1\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-1\right)\left(-2100\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-100±\sqrt{10000+4\left(-2100\right)}}{2\left(-1\right)}$$

Multiply 4 times -2100.

$$x=\frac{-100±\sqrt{10000-8400}}{2\left(-1\right)}$$

Add 10000 to -8400.

$$x=\frac{-100±\sqrt{1600}}{2\left(-1\right)}$$

Take the square root of 1600.

$$x=\frac{-100±40}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-100±40}{-2}$$

Now solve the equation x=\frac{-100±40}{-2} when ± is plus. Add -100 to 40.

$$x=-\frac{60}{-2}$$

Divide -60 by -2.

$$x=30$$

Now solve the equation x=\frac{-100±40}{-2} when ± is minus. Subtract 40 from -100.

$$x=-\frac{140}{-2}$$

Divide -140 by -2.

$$x=70$$

The equation is now solved.

$$x=30 x=70$$

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.

$$-x^{2}+100x-2100=0$$

Add 2100 to both sides of the equation.

$$-x^{2}+100x-2100-\left(-2100\right)=-\left(-2100\right)$$

Subtracting -2100 from itself leaves 0.

$$-x^{2}+100x=-\left(-2100\right)$$

Subtract -2100 from 0.

$$-x^{2}+100x=2100$$

Divide both sides by -1.

$$\frac{-x^{2}+100x}{-1}=\frac{2100}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{100}{-1}x=\frac{2100}{-1}$$

Divide 100 by -1.

$$x^{2}-100x=\frac{2100}{-1}$$

Divide 2100 by -1.

$$x^{2}-100x=-2100$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=-2100+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=-2100+2500$$

Add -2100 to 2500.

$$x^{2}-100x+2500=400$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{400}$$

Simplify.

$$x-50=20 x-50=-20$$

Add 50 to both sides of the equation.

$$x=70 x=30$$