Solve 2(-5/2m-10)=2(12-3/4m^2)

$$2 ( - \frac { 5 } { 2 } m - 10 ) = 2 ( 12 - \frac { 3 } { 4 } m ^ { 2 } )$$

Solve for m

$$m = \frac{22}{3} = 7\frac{1}{3} \approx 7.333333333$$

$$m=-4$$

Steps Using the Quadratic Formula

Cancel out 2 on both sides.

$$-\frac{5}{2}m-10=12-\frac{3}{4}m^{2}$$

Subtract 12 from both sides.

$$-\frac{5}{2}m-10-12=-\frac{3}{4}m^{2}$$

Subtract 12 from -10 to get -22.

$$-\frac{5}{2}m-22=-\frac{3}{4}m^{2}$$

Add \frac{3}{4}m^{2} to both sides.

$$-\frac{5}{2}m-22+\frac{3}{4}m^{2}=0$$

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.

$$\frac{3}{4}m^{2}-\frac{5}{2}m-22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{3}{4} for a, -\frac{5}{2} for b, and -22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-\frac{5}{2}\right)±\sqrt{\left(-\frac{5}{2}\right)^{2}-4\times \frac{3}{4}\left(-22\right)}}{2\times \frac{3}{4}}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$m=\frac{-\left(-\frac{5}{2}\right)±\sqrt{\frac{25}{4}-4\times \frac{3}{4}\left(-22\right)}}{2\times \frac{3}{4}}$$

Multiply -4 times \frac{3}{4}.

$$m=\frac{-\left(-\frac{5}{2}\right)±\sqrt{\frac{25}{4}-3\left(-22\right)}}{2\times \frac{3}{4}}$$

Multiply -3 times -22.

$$m=\frac{-\left(-\frac{5}{2}\right)±\sqrt{\frac{25}{4}+66}}{2\times \frac{3}{4}}$$

Add \frac{25}{4} to 66.

$$m=\frac{-\left(-\frac{5}{2}\right)±\sqrt{\frac{289}{4}}}{2\times \frac{3}{4}}$$

Take the square root of \frac{289}{4}.

$$m=\frac{-\left(-\frac{5}{2}\right)±\frac{17}{2}}{2\times \frac{3}{4}}$$

The opposite of -\frac{5}{2} is \frac{5}{2}.

$$m=\frac{\frac{5}{2}±\frac{17}{2}}{2\times \frac{3}{4}}$$

Multiply 2 times \frac{3}{4}.

$$m=\frac{\frac{5}{2}±\frac{17}{2}}{\frac{3}{2}}$$

Now solve the equation m=\frac{\frac{5}{2}±\frac{17}{2}}{\frac{3}{2}} when ± is plus. Add \frac{5}{2} to \frac{17}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m=\frac{11}{\frac{3}{2}}$$

Divide 11 by \frac{3}{2} by multiplying 11 by the reciprocal of \frac{3}{2}.

$$m=\frac{22}{3}$$

Now solve the equation m=\frac{\frac{5}{2}±\frac{17}{2}}{\frac{3}{2}} when ± is minus. Subtract \frac{17}{2} from \frac{5}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$m=-\frac{6}{\frac{3}{2}}$$

Divide -6 by \frac{3}{2} by multiplying -6 by the reciprocal of \frac{3}{2}.

$$m=-4$$

The equation is now solved.

$$m=\frac{22}{3} m=-4$$

Steps for Completing the Square

Cancel out 2 on both sides.

$$-\frac{5}{2}m-10=12-\frac{3}{4}m^{2}$$

Add \frac{3}{4}m^{2} to both sides.

$$-\frac{5}{2}m-10+\frac{3}{4}m^{2}=12$$

Add 10 to both sides.

$$-\frac{5}{2}m+\frac{3}{4}m^{2}=12+10$$

Add 12 and 10 to get 22.

$$-\frac{5}{2}m+\frac{3}{4}m^{2}=22$$

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.

$$\frac{3}{4}m^{2}-\frac{5}{2}m=22$$

Divide both sides of the equation by \frac{3}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{3}{4}m^{2}-\frac{5}{2}m}{\frac{3}{4}}=\frac{22}{\frac{3}{4}}$$

Dividing by \frac{3}{4} undoes the multiplication by \frac{3}{4}.

$$m^{2}+\left(-\frac{\frac{5}{2}}{\frac{3}{4}}\right)m=\frac{22}{\frac{3}{4}}$$

Divide -\frac{5}{2} by \frac{3}{4} by multiplying -\frac{5}{2} by the reciprocal of \frac{3}{4}.

$$m^{2}-\frac{10}{3}m=\frac{22}{\frac{3}{4}}$$

Divide 22 by \frac{3}{4} by multiplying 22 by the reciprocal of \frac{3}{4}.

$$m^{2}-\frac{10}{3}m=\frac{88}{3}$$

Divide -\frac{10}{3}, the coefficient of the x term, by 2 to get -\frac{5}{3}. Then add the square of -\frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{10}{3}m+\left(-\frac{5}{3}\right)^{2}=\frac{88}{3}+\left(-\frac{5}{3}\right)^{2}$$

Square -\frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{10}{3}m+\frac{25}{9}=\frac{88}{3}+\frac{25}{9}$$

Add \frac{88}{3} to \frac{25}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}-\frac{10}{3}m+\frac{25}{9}=\frac{289}{9}$$

Factor m^{2}-\frac{10}{3}m+\frac{25}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-\frac{5}{3}\right)^{2}=\frac{289}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{5}{3}\right)^{2}}=\sqrt{\frac{289}{9}}$$

Simplify.

$$m-\frac{5}{3}=\frac{17}{3} m-\frac{5}{3}=-\frac{17}{3}$$

Add \frac{5}{3} to both sides of the equation.

$$m=\frac{22}{3} m=-4$$