Solve m^2+75=100m

$$m ^ { 2 } + 75 = 100 m$$

Solve for m

$$m=5\sqrt{97}+50\approx 99.244289009$$

$$m=50-5\sqrt{97}\approx 0.755710991$$

Steps Using the Quadratic Formula

Subtract 100m from both sides.

$$m^{2}+75-100m=0$$

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.

$$m^{2}-100m+75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -100 for b, and 75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-100\right)±\sqrt{\left(-100\right)^{2}-4\times 75}}{2}$$

Square -100.

$$m=\frac{-\left(-100\right)±\sqrt{10000-4\times 75}}{2}$$

Multiply -4 times 75.

$$m=\frac{-\left(-100\right)±\sqrt{10000-300}}{2}$$

Add 10000 to -300.

$$m=\frac{-\left(-100\right)±\sqrt{9700}}{2}$$

Take the square root of 9700.

$$m=\frac{-\left(-100\right)±10\sqrt{97}}{2}$$

The opposite of -100 is 100.

$$m=\frac{100±10\sqrt{97}}{2}$$

Now solve the equation m=\frac{100±10\sqrt{97}}{2} when ± is plus. Add 100 to 10\sqrt{97}.

$$m=\frac{10\sqrt{97}+100}{2}$$

Divide 100+10\sqrt{97} by 2.

$$m=5\sqrt{97}+50$$

Now solve the equation m=\frac{100±10\sqrt{97}}{2} when ± is minus. Subtract 10\sqrt{97} from 100.

$$m=\frac{100-10\sqrt{97}}{2}$$

Divide 100-10\sqrt{97} by 2.

$$m=50-5\sqrt{97}$$

The equation is now solved.

$$m=5\sqrt{97}+50 m=50-5\sqrt{97}$$

Steps for Completing the Square

Subtract 100m from both sides.

$$m^{2}+75-100m=0$$

Subtract 75 from both sides. Anything subtracted from zero gives its negation.

$$m^{2}-100m=-75$$

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.

$$m^{2}-100m+\left(-50\right)^{2}=-75+\left(-50\right)^{2}$$

Square -50.

$$m^{2}-100m+2500=-75+2500$$

Add -75 to 2500.

$$m^{2}-100m+2500=2425$$

Factor m^{2}-100m+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}.

$$\left(m-50\right)^{2}=2425$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-50\right)^{2}}=\sqrt{2425}$$

Simplify.

$$m-50=5\sqrt{97} m-50=-5\sqrt{97}$$

Add 50 to both sides of the equation.

$$m=5\sqrt{97}+50 m=50-5\sqrt{97}$$