Solve (i+1)2i^2+i+6i+6/6

$$( i + 1 ) \frac { 2 i ^ { 2 } + i + 6 i + 6 } { 6 }$$

Evaluate

$$-\frac{1}{2}+\frac{11}{6}i\approx -0.5+1.833333333i$$

Solution Steps

Do the additions in i+6i+6.

$$\left(i+1\right)\times \frac{2i^{2}+6+7i}{6}$$

Calculate i to the power of 2 and get -1.

$$\left(i+1\right)\times \frac{2\left(-1\right)+6+7i}{6}$$

Multiply 2 and -1 to get -2.

$$\left(i+1\right)\times \frac{-2+6+7i}{6}$$

Do the additions in -2+6+7i.

$$\left(i+1\right)\times \frac{4+7i}{6}$$

Divide 4+7i by 6 to get \frac{2}{3}+\frac{7}{6}i.

$$\left(i+1\right)\left(\frac{2}{3}+\frac{7}{6}i\right)$$

Use the distributive property to multiply i+1 by \frac{2}{3}+\frac{7}{6}i.

$$-\frac{7}{6}+\frac{2}{3}i+\left(\frac{2}{3}+\frac{7}{6}i\right)$$

Add -\frac{7}{6}+\frac{2}{3}i and \frac{2}{3}+\frac{7}{6}i to get -\frac{1}{2}+\frac{11}{6}i.

$$-\frac{1}{2}+\frac{11}{6}i$$

Real Part

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

Solution Steps

Do the additions in i+6i+6.

$$Re(\left(i+1\right)\times \frac{2i^{2}+6+7i}{6})$$

Calculate i to the power of 2 and get -1.

$$Re(\left(i+1\right)\times \frac{2\left(-1\right)+6+7i}{6})$$

Multiply 2 and -1 to get -2.

$$Re(\left(i+1\right)\times \frac{-2+6+7i}{6})$$

Do the additions in -2+6+7i.

$$Re(\left(i+1\right)\times \frac{4+7i}{6})$$

Divide 4+7i by 6 to get \frac{2}{3}+\frac{7}{6}i.

$$Re(\left(i+1\right)\left(\frac{2}{3}+\frac{7}{6}i\right))$$

Use the distributive property to multiply i+1 by \frac{2}{3}+\frac{7}{6}i.

$$Re(-\frac{7}{6}+\frac{2}{3}i+\left(\frac{2}{3}+\frac{7}{6}i\right))$$

Add -\frac{7}{6}+\frac{2}{3}i and \frac{2}{3}+\frac{7}{6}i to get -\frac{1}{2}+\frac{11}{6}i.

$$Re(-\frac{1}{2}+\frac{11}{6}i)$$

The real part of -\frac{1}{2}+\frac{11}{6}i is -\frac{1}{2}.

$$-\frac{1}{2}$$