Solve 2500a-100a^2=10000

$$2500 a - 100 a ^ { 2 } = 10000$$

Solve for a

$$a=5$$

$$a=20$$

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.

$$-100a^{2}+2500a=10000$$

Subtract 10000 from both sides of the equation.

$$-100a^{2}+2500a-10000=10000-10000$$

Subtracting 10000 from itself leaves 0.

$$-100a^{2}+2500a-10000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -100 for a, 2500 for b, and -10000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-2500±\sqrt{2500^{2}-4\left(-100\right)\left(-10000\right)}}{2\left(-100\right)}$$

Square 2500.

$$a=\frac{-2500±\sqrt{6250000-4\left(-100\right)\left(-10000\right)}}{2\left(-100\right)}$$

Multiply -4 times -100.

$$a=\frac{-2500±\sqrt{6250000+400\left(-10000\right)}}{2\left(-100\right)}$$

Multiply 400 times -10000.

$$a=\frac{-2500±\sqrt{6250000-4000000}}{2\left(-100\right)}$$

Add 6250000 to -4000000.

$$a=\frac{-2500±\sqrt{2250000}}{2\left(-100\right)}$$

Take the square root of 2250000.

$$a=\frac{-2500±1500}{2\left(-100\right)}$$

Multiply 2 times -100.

$$a=\frac{-2500±1500}{-200}$$

Now solve the equation a=\frac{-2500±1500}{-200} when ± is plus. Add -2500 to 1500.

$$a=-\frac{1000}{-200}$$

Divide -1000 by -200.

$$a=5$$

Now solve the equation a=\frac{-2500±1500}{-200} when ± is minus. Subtract 1500 from -2500.

$$a=-\frac{4000}{-200}$$

Divide -4000 by -200.

$$a=20$$

The equation is now solved.

$$a=5 a=20$$

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.

$$-100a^{2}+2500a=10000$$

Divide both sides by -100.

$$\frac{-100a^{2}+2500a}{-100}=\frac{10000}{-100}$$

Dividing by -100 undoes the multiplication by -100.

$$a^{2}+\frac{2500}{-100}a=\frac{10000}{-100}$$

Divide 2500 by -100.

$$a^{2}-25a=\frac{10000}{-100}$$

Divide 10000 by -100.

$$a^{2}-25a=-100$$

Divide -25, the coefficient of the x term, by 2 to get -\frac{25}{2}. Then add the square of -\frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-25a+\left(-\frac{25}{2}\right)^{2}=-100+\left(-\frac{25}{2}\right)^{2}$$

Square -\frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-25a+\frac{625}{4}=-100+\frac{625}{4}$$

Add -100 to \frac{625}{4}.

$$a^{2}-25a+\frac{625}{4}=\frac{225}{4}$$

Factor a^{2}-25a+\frac{625}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{25}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{25}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$a-\frac{25}{2}=\frac{15}{2} a-\frac{25}{2}=-\frac{15}{2}$$

Add \frac{25}{2} to both sides of the equation.

$$a=20 a=5$$