Solve 3/4m^2=(3/2m+3)(2-m)/2+3/2

$$\frac { 3 } { 4 } m ^ { 2 } = \frac { ( \frac { 3 } { 2 } m + 3 ) ( 2 - m ) } { 2 } + \frac { 3 } { 2 }$$

Solve for m

$$m=\sqrt{3}\approx 1.732050808$$

$$m=-\sqrt{3}\approx -1.732050808$$

Steps by Finding Square Root

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$3m^{2}=2\left(\frac{3}{2}m+3\right)\left(2-m\right)+6$$

Use the distributive property to multiply 2 by \frac{3}{2}m+3.

$$3m^{2}=\left(3m+6\right)\left(2-m\right)+6$$

Use the distributive property to multiply 3m+6 by 2-m and combine like terms.

$$3m^{2}=-3m^{2}+12+6$$

Add 12 and 6 to get 18.

$$3m^{2}=-3m^{2}+18$$

Add 3m^{2} to both sides.

$$3m^{2}+3m^{2}=18$$

Combine 3m^{2} and 3m^{2} to get 6m^{2}.

$$6m^{2}=18$$

Divide both sides by 6.

$$m^{2}=\frac{18}{6}$$

Divide 18 by 6 to get 3.

$$m^{2}=3$$

Take the square root of both sides of the equation.

$$m=\sqrt{3} m=-\sqrt{3}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$3m^{2}=2\left(\frac{3}{2}m+3\right)\left(2-m\right)+6$$

Use the distributive property to multiply 2 by \frac{3}{2}m+3.

$$3m^{2}=\left(3m+6\right)\left(2-m\right)+6$$

Use the distributive property to multiply 3m+6 by 2-m and combine like terms.

$$3m^{2}=-3m^{2}+12+6$$

Add 12 and 6 to get 18.

$$3m^{2}=-3m^{2}+18$$

Add 3m^{2} to both sides.

$$3m^{2}+3m^{2}=18$$

Combine 3m^{2} and 3m^{2} to get 6m^{2}.

$$6m^{2}=18$$

Subtract 18 from both sides.

$$6m^{2}-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 0 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{0±\sqrt{0^{2}-4\times 6\left(-18\right)}}{2\times 6}$$

Square 0.

$$m=\frac{0±\sqrt{-4\times 6\left(-18\right)}}{2\times 6}$$

Multiply -4 times 6.

$$m=\frac{0±\sqrt{-24\left(-18\right)}}{2\times 6}$$

Multiply -24 times -18.

$$m=\frac{0±\sqrt{432}}{2\times 6}$$

Take the square root of 432.

$$m=\frac{0±12\sqrt{3}}{2\times 6}$$

Multiply 2 times 6.

$$m=\frac{0±12\sqrt{3}}{12}$$

Now solve the equation m=\frac{0±12\sqrt{3}}{12} when ± is plus.

$$m=\sqrt{3}$$

Now solve the equation m=\frac{0±12\sqrt{3}}{12} when ± is minus.

$$m=-\sqrt{3}$$

The equation is now solved.

$$m=\sqrt{3} m=-\sqrt{3}$$