Solve 1/2(8-2(m+1)^2+10-(m+1)^2)=75/8

$$\frac { 1 } { 2 } ( 8 - 2 ( m + 1 ) ^ { 2 } + 10 - ( m + 1 ) ^ { 2 } ) = \frac { 75 } { 8 }$$

Solve for m

$$m=-1-\frac{1}{2}i=-1-0.5i$$

$$m=-1+\frac{1}{2}i=-1+0.5i$$

Steps Using the Quadratic Formula

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$8-2\left(m+1\right)^{2}+10-\left(m+1\right)^{2}=\frac{75}{8}\times 2$$

Multiply \frac{75}{8} and 2 to get \frac{75}{4}.

$$8-2\left(m+1\right)^{2}+10-\left(m+1\right)^{2}=\frac{75}{4}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+1\right)^{2}.

$$8-2\left(m^{2}+2m+1\right)+10-\left(m+1\right)^{2}=\frac{75}{4}$$

Use the distributive property to multiply -2 by m^{2}+2m+1.

$$8-2m^{2}-4m-2+10-\left(m+1\right)^{2}=\frac{75}{4}$$

Subtract 2 from 8 to get 6.

$$6-2m^{2}-4m+10-\left(m+1\right)^{2}=\frac{75}{4}$$

Add 6 and 10 to get 16.

$$16-2m^{2}-4m-\left(m+1\right)^{2}=\frac{75}{4}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+1\right)^{2}.

$$16-2m^{2}-4m-\left(m^{2}+2m+1\right)=\frac{75}{4}$$

To find the opposite of m^{2}+2m+1, find the opposite of each term.

$$16-2m^{2}-4m-m^{2}-2m-1=\frac{75}{4}$$

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

$$16-3m^{2}-4m-2m-1=\frac{75}{4}$$

Combine -4m and -2m to get -6m.

$$16-3m^{2}-6m-1=\frac{75}{4}$$

Subtract 1 from 16 to get 15.

$$15-3m^{2}-6m=\frac{75}{4}$$

Subtract \frac{75}{4} from both sides.

$$15-3m^{2}-6m-\frac{75}{4}=0$$

Subtract \frac{75}{4} from 15 to get -\frac{15}{4}.

$$-\frac{15}{4}-3m^{2}-6m=0$$

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.

$$-3m^{2}-6m-\frac{15}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -6 for b, and -\frac{15}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-3\right)\left(-\frac{15}{4}\right)}}{2\left(-3\right)}$$

Square -6.

$$m=\frac{-\left(-6\right)±\sqrt{36-4\left(-3\right)\left(-\frac{15}{4}\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$m=\frac{-\left(-6\right)±\sqrt{36+12\left(-\frac{15}{4}\right)}}{2\left(-3\right)}$$

Multiply 12 times -\frac{15}{4}.

$$m=\frac{-\left(-6\right)±\sqrt{36-45}}{2\left(-3\right)}$$

Add 36 to -45.

$$m=\frac{-\left(-6\right)±\sqrt{-9}}{2\left(-3\right)}$$

Take the square root of -9.

$$m=\frac{-\left(-6\right)±3i}{2\left(-3\right)}$$

The opposite of -6 is 6.

$$m=\frac{6±3i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$m=\frac{6±3i}{-6}$$

Now solve the equation m=\frac{6±3i}{-6} when ± is plus. Add 6 to 3i.

$$m=\frac{6+3i}{-6}$$

Divide 6+3i by -6.

$$m=-1-\frac{1}{2}i$$

Now solve the equation m=\frac{6±3i}{-6} when ± is minus. Subtract 3i from 6.

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

Divide 6-3i by -6.

$$m=-1+\frac{1}{2}i$$

The equation is now solved.

$$m=-1-\frac{1}{2}i m=-1+\frac{1}{2}i$$

Steps for Completing the Square

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$8-2\left(m+1\right)^{2}+10-\left(m+1\right)^{2}=\frac{75}{8}\times 2$$

Multiply \frac{75}{8} and 2 to get \frac{75}{4}.

$$8-2\left(m+1\right)^{2}+10-\left(m+1\right)^{2}=\frac{75}{4}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+1\right)^{2}.

$$8-2\left(m^{2}+2m+1\right)+10-\left(m+1\right)^{2}=\frac{75}{4}$$

Use the distributive property to multiply -2 by m^{2}+2m+1.

$$8-2m^{2}-4m-2+10-\left(m+1\right)^{2}=\frac{75}{4}$$

Subtract 2 from 8 to get 6.

$$6-2m^{2}-4m+10-\left(m+1\right)^{2}=\frac{75}{4}$$

Add 6 and 10 to get 16.

$$16-2m^{2}-4m-\left(m+1\right)^{2}=\frac{75}{4}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+1\right)^{2}.

$$16-2m^{2}-4m-\left(m^{2}+2m+1\right)=\frac{75}{4}$$

To find the opposite of m^{2}+2m+1, find the opposite of each term.

$$16-2m^{2}-4m-m^{2}-2m-1=\frac{75}{4}$$

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

$$16-3m^{2}-4m-2m-1=\frac{75}{4}$$

Combine -4m and -2m to get -6m.

$$16-3m^{2}-6m-1=\frac{75}{4}$$

Subtract 1 from 16 to get 15.

$$15-3m^{2}-6m=\frac{75}{4}$$

Subtract 15 from both sides.

$$-3m^{2}-6m=\frac{75}{4}-15$$

Subtract 15 from \frac{75}{4} to get \frac{15}{4}.

$$-3m^{2}-6m=\frac{15}{4}$$

Divide both sides by -3.

$$\frac{-3m^{2}-6m}{-3}=\frac{\frac{15}{4}}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$m^{2}+\left(-\frac{6}{-3}\right)m=\frac{\frac{15}{4}}{-3}$$

Divide -6 by -3.

$$m^{2}+2m=\frac{\frac{15}{4}}{-3}$$

Divide \frac{15}{4} by -3.

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

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+2m+1^{2}=-\frac{5}{4}+1^{2}$$

Square 1.

$$m^{2}+2m+1=-\frac{5}{4}+1$$

Add -\frac{5}{4} to 1.

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

Factor m^{2}+2m+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$m+1=\frac{1}{2}i m+1=-\frac{1}{2}i$$

Subtract 1 from both sides of the equation.

$$m=-1+\frac{1}{2}i m=-1-\frac{1}{2}i$$