Solve 10m^2-996m+39600=0

$$10 { m }^{ 2 } -996m+39600 = 0$$

Solve for m

$$m=\frac{249+3\sqrt{4111}i}{5}\approx 49.8+38.470248245i$$

$$m=\frac{-3\sqrt{4111}i+249}{5}\approx 49.8-38.470248245i$$

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.

$$10m^{2}-996m+39600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, -996 for b, and 39600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-996\right)±\sqrt{\left(-996\right)^{2}-4\times 10\times 39600}}{2\times 10}$$

Square -996.

$$m=\frac{-\left(-996\right)±\sqrt{992016-4\times 10\times 39600}}{2\times 10}$$

Multiply -4 times 10.

$$m=\frac{-\left(-996\right)±\sqrt{992016-40\times 39600}}{2\times 10}$$

Multiply -40 times 39600.

$$m=\frac{-\left(-996\right)±\sqrt{992016-1584000}}{2\times 10}$$

Add 992016 to -1584000.

$$m=\frac{-\left(-996\right)±\sqrt{-591984}}{2\times 10}$$

Take the square root of -591984.

$$m=\frac{-\left(-996\right)±12\sqrt{4111}i}{2\times 10}$$

The opposite of -996 is 996.

$$m=\frac{996±12\sqrt{4111}i}{2\times 10}$$

Multiply 2 times 10.

$$m=\frac{996±12\sqrt{4111}i}{20}$$

Now solve the equation m=\frac{996±12\sqrt{4111}i}{20} when ± is plus. Add 996 to 12i\sqrt{4111}.

$$m=\frac{996+12\sqrt{4111}i}{20}$$

Divide 996+12i\sqrt{4111} by 20.

$$m=\frac{249+3\sqrt{4111}i}{5}$$

Now solve the equation m=\frac{996±12\sqrt{4111}i}{20} when ± is minus. Subtract 12i\sqrt{4111} from 996.

$$m=\frac{-12\sqrt{4111}i+996}{20}$$

Divide 996-12i\sqrt{4111} by 20.

$$m=\frac{-3\sqrt{4111}i+249}{5}$$

The equation is now solved.

$$m=\frac{249+3\sqrt{4111}i}{5} m=\frac{-3\sqrt{4111}i+249}{5}$$

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.

$$10m^{2}-996m+39600=0$$

Subtract 39600 from both sides of the equation.

$$10m^{2}-996m+39600-39600=-39600$$

Subtracting 39600 from itself leaves 0.

$$10m^{2}-996m=-39600$$

Divide both sides by 10.

$$\frac{10m^{2}-996m}{10}=-\frac{39600}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$m^{2}+\left(-\frac{996}{10}\right)m=-\frac{39600}{10}$$

Reduce the fraction \frac{-996}{10} to lowest terms by extracting and canceling out 2.

$$m^{2}-\frac{498}{5}m=-\frac{39600}{10}$$

Divide -39600 by 10.

$$m^{2}-\frac{498}{5}m=-3960$$

Divide -\frac{498}{5}, the coefficient of the x term, by 2 to get -\frac{249}{5}. Then add the square of -\frac{249}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{498}{5}m+\left(-\frac{249}{5}\right)^{2}=-3960+\left(-\frac{249}{5}\right)^{2}$$

Square -\frac{249}{5} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{498}{5}m+\frac{62001}{25}=-3960+\frac{62001}{25}$$

Add -3960 to \frac{62001}{25}.

$$m^{2}-\frac{498}{5}m+\frac{62001}{25}=-\frac{36999}{25}$$

Factor m^{2}-\frac{498}{5}m+\frac{62001}{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}.

$$\left(m-\frac{249}{5}\right)^{2}=-\frac{36999}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{249}{5}\right)^{2}}=\sqrt{-\frac{36999}{25}}$$

Simplify.

$$m-\frac{249}{5}=\frac{3\sqrt{4111}i}{5} m-\frac{249}{5}=-\frac{3\sqrt{4111}i}{5}$$

Add \frac{249}{5} to both sides of the equation.

$$m=\frac{249+3\sqrt{4111}i}{5} m=\frac{-3\sqrt{4111}i+249}{5}$$