Solve 23m^2+23-50m=0

$$23 m ^ { 2 } + 23 - 50 m = 0$$

Solve for m

$$m = \frac{4 \sqrt{6} + 25}{23} \approx 1.512954738$$

$$m=\frac{25-4\sqrt{6}}{23}\approx 0.660958306$$

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.

$$23m^{2}-50m+23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 23 for a, -50 for b, and 23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 23\times 23}}{2\times 23}$$

Square -50.

$$m=\frac{-\left(-50\right)±\sqrt{2500-4\times 23\times 23}}{2\times 23}$$

Multiply -4 times 23.

$$m=\frac{-\left(-50\right)±\sqrt{2500-92\times 23}}{2\times 23}$$

Multiply -92 times 23.

$$m=\frac{-\left(-50\right)±\sqrt{2500-2116}}{2\times 23}$$

Add 2500 to -2116.

$$m=\frac{-\left(-50\right)±\sqrt{384}}{2\times 23}$$

Take the square root of 384.

$$m=\frac{-\left(-50\right)±8\sqrt{6}}{2\times 23}$$

The opposite of -50 is 50.

$$m=\frac{50±8\sqrt{6}}{2\times 23}$$

Multiply 2 times 23.

$$m=\frac{50±8\sqrt{6}}{46}$$

Now solve the equation m=\frac{50±8\sqrt{6}}{46} when ± is plus. Add 50 to 8\sqrt{6}.

$$m=\frac{8\sqrt{6}+50}{46}$$

Divide 50+8\sqrt{6} by 46.

$$m=\frac{4\sqrt{6}+25}{23}$$

Now solve the equation m=\frac{50±8\sqrt{6}}{46} when ± is minus. Subtract 8\sqrt{6} from 50.

$$m=\frac{50-8\sqrt{6}}{46}$$

Divide 50-8\sqrt{6} by 46.

$$m=\frac{25-4\sqrt{6}}{23}$$

The equation is now solved.

$$m=\frac{4\sqrt{6}+25}{23} m=\frac{25-4\sqrt{6}}{23}$$

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.

$$23m^{2}-50m+23=0$$

Subtract 23 from both sides of the equation.

$$23m^{2}-50m+23-23=-23$$

Subtracting 23 from itself leaves 0.

$$23m^{2}-50m=-23$$

Divide both sides by 23.

$$\frac{23m^{2}-50m}{23}=-\frac{23}{23}$$

Dividing by 23 undoes the multiplication by 23.

$$m^{2}-\frac{50}{23}m=-\frac{23}{23}$$

Divide -23 by 23.

$$m^{2}-\frac{50}{23}m=-1$$

Divide -\frac{50}{23}, the coefficient of the x term, by 2 to get -\frac{25}{23}. Then add the square of -\frac{25}{23} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{50}{23}m+\left(-\frac{25}{23}\right)^{2}=-1+\left(-\frac{25}{23}\right)^{2}$$

Square -\frac{25}{23} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{50}{23}m+\frac{625}{529}=-1+\frac{625}{529}$$

Add -1 to \frac{625}{529}.

$$m^{2}-\frac{50}{23}m+\frac{625}{529}=\frac{96}{529}$$

Factor m^{2}-\frac{50}{23}m+\frac{625}{529}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-\frac{25}{23}\right)^{2}=\frac{96}{529}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{25}{23}\right)^{2}}=\sqrt{\frac{96}{529}}$$

Simplify.

$$m-\frac{25}{23}=\frac{4\sqrt{6}}{23} m-\frac{25}{23}=-\frac{4\sqrt{6}}{23}$$

Add \frac{25}{23} to both sides of the equation.

$$m=\frac{4\sqrt{6}+25}{23} m=\frac{25-4\sqrt{6}}{23}$$