Solve -500a^2+53000a=600000

$$-500 { a }^{ 2 } +53000a = 600000$$

Solve for a

$$a=\sqrt{1609}+53\approx 93.11234224$$

$$a=53-\sqrt{1609}\approx 12.88765776$$

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.

$$-500a^{2}+53000a=600000$$

Subtract 600000 from both sides of the equation.

$$-500a^{2}+53000a-600000=600000-600000$$

Subtracting 600000 from itself leaves 0.

$$-500a^{2}+53000a-600000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -500 for a, 53000 for b, and -600000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-53000±\sqrt{53000^{2}-4\left(-500\right)\left(-600000\right)}}{2\left(-500\right)}$$

Square 53000.

$$a=\frac{-53000±\sqrt{2809000000-4\left(-500\right)\left(-600000\right)}}{2\left(-500\right)}$$

Multiply -4 times -500.

$$a=\frac{-53000±\sqrt{2809000000+2000\left(-600000\right)}}{2\left(-500\right)}$$

Multiply 2000 times -600000.

$$a=\frac{-53000±\sqrt{2809000000-1200000000}}{2\left(-500\right)}$$

Add 2809000000 to -1200000000.

$$a=\frac{-53000±\sqrt{1609000000}}{2\left(-500\right)}$$

Take the square root of 1609000000.

$$a=\frac{-53000±1000\sqrt{1609}}{2\left(-500\right)}$$

Multiply 2 times -500.

$$a=\frac{-53000±1000\sqrt{1609}}{-1000}$$

Now solve the equation a=\frac{-53000±1000\sqrt{1609}}{-1000} when ± is plus. Add -53000 to 1000\sqrt{1609}.

$$a=\frac{1000\sqrt{1609}-53000}{-1000}$$

Divide -53000+1000\sqrt{1609} by -1000.

$$a=53-\sqrt{1609}$$

Now solve the equation a=\frac{-53000±1000\sqrt{1609}}{-1000} when ± is minus. Subtract 1000\sqrt{1609} from -53000.

$$a=\frac{-1000\sqrt{1609}-53000}{-1000}$$

Divide -53000-1000\sqrt{1609} by -1000.

$$a=\sqrt{1609}+53$$

The equation is now solved.

$$a=53-\sqrt{1609} a=\sqrt{1609}+53$$

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.

$$-500a^{2}+53000a=600000$$

Divide both sides by -500.

$$\frac{-500a^{2}+53000a}{-500}=\frac{600000}{-500}$$

Dividing by -500 undoes the multiplication by -500.

$$a^{2}+\frac{53000}{-500}a=\frac{600000}{-500}$$

Divide 53000 by -500.

$$a^{2}-106a=\frac{600000}{-500}$$

Divide 600000 by -500.

$$a^{2}-106a=-1200$$

Divide -106, the coefficient of the x term, by 2 to get -53. Then add the square of -53 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-106a+\left(-53\right)^{2}=-1200+\left(-53\right)^{2}$$

Square -53.

$$a^{2}-106a+2809=-1200+2809$$

Add -1200 to 2809.

$$a^{2}-106a+2809=1609$$

Factor a^{2}-106a+2809. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-53\right)^{2}=1609$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-53\right)^{2}}=\sqrt{1609}$$

Simplify.

$$a-53=\sqrt{1609} a-53=-\sqrt{1609}$$

Add 53 to both sides of the equation.

$$a=\sqrt{1609}+53 a=53-\sqrt{1609}$$