Solve (a+ib-5)(a-ib-5

$$( a + i b - 5 ) ( a - i b - 5$$

Evaluate

$$\left(a-5\right)^{2}+b^{2}$$

Solution Steps

Apply the distributive property by multiplying each term of a+ib-5 by each term of a-ib-5.

$$a\left(a-ib\right)-5a+ib\left(a-ib\right)-5ib-5\left(a-ib\right)+25$$

Use the distributive property to multiply a by a-ib.

$$a^{2}-iab-5a+ib\left(a-ib\right)-5ib-5\left(a-ib\right)+25$$

Use the distributive property to multiply ib by a-ib.

$$a^{2}-iab-5a+iba+b^{2}-5ib-5\left(a-ib\right)+25$$

Combine -iab and iba to get 0.

$$a^{2}-5a+b^{2}-5ib-5\left(a-ib\right)+25$$

Use the distributive property to multiply -5 by a-ib.

$$a^{2}-5a+b^{2}-5ib-5a+5ib+25$$

Combine -5a and -5a to get -10a.

$$a^{2}-10a+b^{2}-5ib+5ib+25$$

Combine -5ib and 5ib to get 0.

$$a^{2}-10a+b^{2}+25$$

Expand

$$a^{2}-10a+b^{2}+25$$

Solution Steps

Apply the distributive property by multiplying each term of a+ib-5 by each term of a-ib-5.

$$a\left(a-ib\right)-5a+ib\left(a-ib\right)-5ib-5\left(a-ib\right)+25$$

Use the distributive property to multiply a by a-ib.

$$a^{2}-iab-5a+ib\left(a-ib\right)-5ib-5\left(a-ib\right)+25$$

Use the distributive property to multiply ib by a-ib.

$$a^{2}-iab-5a+iba+b^{2}-5ib-5\left(a-ib\right)+25$$

Combine -iab and iba to get 0.

$$a^{2}-5a+b^{2}-5ib-5\left(a-ib\right)+25$$

Use the distributive property to multiply -5 by a-ib.

$$a^{2}-5a+b^{2}-5ib-5a+5ib+25$$

Combine -5a and -5a to get -10a.

$$a^{2}-10a+b^{2}-5ib+5ib+25$$

Combine -5ib and 5ib to get 0.

$$a^{2}-10a+b^{2}+25$$