Solve left(6+1/2mright)left(6+mright)+8left(6-1/4mright)=6left(6+8right)+11m-8

$$\left( 6+ \frac{ 1 }{ 2 } m \right) \left( 6+m \right) +8 \left( 6- \frac{ 1 }{ 4 } m \right) = 6 \left( 6+8 \right) +11m-8$$

Solve for m

$$m=4$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6+\frac{1}{2}m by 6+m and combine like terms.

$$36+9m+\frac{1}{2}m^{2}+8\left(6-\frac{1}{4}m\right)=6\left(6+8\right)+11m-8$$

Use the distributive property to multiply 8 by 6-\frac{1}{4}m.

$$36+9m+\frac{1}{2}m^{2}+48-2m=6\left(6+8\right)+11m-8$$

Add 36 and 48 to get 84.

$$84+9m+\frac{1}{2}m^{2}-2m=6\left(6+8\right)+11m-8$$

Combine 9m and -2m to get 7m.

$$84+7m+\frac{1}{2}m^{2}=6\left(6+8\right)+11m-8$$

Add 6 and 8 to get 14.

$$84+7m+\frac{1}{2}m^{2}=6\times 14+11m-8$$

Multiply 6 and 14 to get 84.

$$84+7m+\frac{1}{2}m^{2}=84+11m-8$$

Subtract 8 from 84 to get 76.

$$84+7m+\frac{1}{2}m^{2}=76+11m$$

Subtract 76 from both sides.

$$84+7m+\frac{1}{2}m^{2}-76=11m$$

Subtract 76 from 84 to get 8.

$$8+7m+\frac{1}{2}m^{2}=11m$$

Subtract 11m from both sides.

$$8+7m+\frac{1}{2}m^{2}-11m=0$$

Combine 7m and -11m to get -4m.

$$8-4m+\frac{1}{2}m^{2}=0$$

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.

$$\frac{1}{2}m^{2}-4m+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, -4 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times \frac{1}{2}\times 8}}{2\times \frac{1}{2}}$$

Square -4.

$$m=\frac{-\left(-4\right)±\sqrt{16-4\times \frac{1}{2}\times 8}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$m=\frac{-\left(-4\right)±\sqrt{16-2\times 8}}{2\times \frac{1}{2}}$$

Multiply -2 times 8.

$$m=\frac{-\left(-4\right)±\sqrt{16-16}}{2\times \frac{1}{2}}$$

Add 16 to -16.

$$m=\frac{-\left(-4\right)±\sqrt{0}}{2\times \frac{1}{2}}$$

Take the square root of 0.

$$m=-\frac{-4}{2\times \frac{1}{2}}$$

The opposite of -4 is 4.

$$m=\frac{4}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$m=\frac{4}{1}$$

Steps for Completing the Square

Use the distributive property to multiply 6+\frac{1}{2}m by 6+m and combine like terms.

$$36+9m+\frac{1}{2}m^{2}+8\left(6-\frac{1}{4}m\right)=6\left(6+8\right)+11m-8$$

Use the distributive property to multiply 8 by 6-\frac{1}{4}m.

$$36+9m+\frac{1}{2}m^{2}+48-2m=6\left(6+8\right)+11m-8$$

Add 36 and 48 to get 84.

$$84+9m+\frac{1}{2}m^{2}-2m=6\left(6+8\right)+11m-8$$

Combine 9m and -2m to get 7m.

$$84+7m+\frac{1}{2}m^{2}=6\left(6+8\right)+11m-8$$

Add 6 and 8 to get 14.

$$84+7m+\frac{1}{2}m^{2}=6\times 14+11m-8$$

Multiply 6 and 14 to get 84.

$$84+7m+\frac{1}{2}m^{2}=84+11m-8$$

Subtract 8 from 84 to get 76.

$$84+7m+\frac{1}{2}m^{2}=76+11m$$

Subtract 11m from both sides.

$$84+7m+\frac{1}{2}m^{2}-11m=76$$

Combine 7m and -11m to get -4m.

$$84-4m+\frac{1}{2}m^{2}=76$$

Subtract 84 from both sides.

$$-4m+\frac{1}{2}m^{2}=76-84$$

Subtract 84 from 76 to get -8.

$$-4m+\frac{1}{2}m^{2}=-8$$

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.

$$\frac{1}{2}m^{2}-4m=-8$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}m^{2}-4m}{\frac{1}{2}}=-\frac{8}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$m^{2}+\left(-\frac{4}{\frac{1}{2}}\right)m=-\frac{8}{\frac{1}{2}}$$

Divide -4 by \frac{1}{2} by multiplying -4 by the reciprocal of \frac{1}{2}.

$$m^{2}-8m=-\frac{8}{\frac{1}{2}}$$

Divide -8 by \frac{1}{2} by multiplying -8 by the reciprocal of \frac{1}{2}.

$$m^{2}-8m=-16$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-8m+\left(-4\right)^{2}=-16+\left(-4\right)^{2}$$

Square -4.

$$m^{2}-8m+16=-16+16$$

Add -16 to 16.

$$m^{2}-8m+16=0$$

Factor m^{2}-8m+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-4\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-4\right)^{2}}=\sqrt{0}$$

Simplify.

$$m-4=0 m-4=0$$

Add 4 to both sides of the equation.

$$m=4 m=4$$

The equation is now solved. Solutions are the same.

$$m=4$$