The product of (3 + 2i) and a complex number is (17 + 7i). What is the complex number?

Solution:

Given, the product of (3 + 2i) and a complex number is (17 + 7i).

We have to find the complex number.

Let the complex number be a.

Now, (3 + 2i) × a = (17 + 7i)

a = (17 + 7i)/(3 + 2i)

On taking conjugate,

a = [(17 + 7i)/(3 + 2i)] × [(3 - 2i)/(3 - 2i)]

a = [(17 + 7i)(3 - 2i)] / [(3 + 2i)(3 - 2i)]

Considering (17 + 7i)(3 - 2i),

(17 + 7i)(3 - 2i) = 17(3) + 17(-2i) + 7i(3) + 7i(-2i)

= 51 - 34i + 21i - 14i²

We know, i² = -1

= 51 - 13i - 14(-1)

= 51 - 13i + 14

= 65 - 13i

Considering (3 + 2i)(3 - 2i),

We know, (a + b)(a - b) = a² - b²

(3 + 2i)(3 - 2i) = (3)² - (2i)²

= 9 - 4i²

= 9 - 4(-1)

= 9 + 4

= 13

Now, a = 65 - 13i / 13

= 13(5 - i) / 13

= 5 - i

Therefore, the complex number is 5 - i.

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The product of (3 + 2i) and a complex number is (17 + 7i). What is the complex number?

Summary:

The product of (3 + 2i) and a complex number is (17 + 7i). The complex number is 5 - i.