Question

If (2 + 3i) (5 - 3i) (-4 - 7i) =


x + 2yi , find the value of x and y

Answer 1

Question :–

${ \huge {.}} { \bold{ If \: \:(2 + 3 \iota) (5 - 3 \iota ) (-4 - 7 \iota) = x + (2y) \iota}} \: \:$ then find the value of x and y .

ANSWER :–

GIVEN :–

▪︎ $\: \: { \bold{ (2 + 3 \iota) (5 - 3 \iota ) (-4 - 7 \iota) = x + (2y) \iota}} \: \:$

TO FIND :–

▪︎ Value of x and y.

SOLUTION :–

$\\ \implies { \bold{ (2 + 3\iota) (5 - 3\iota) (-4 - 7\iota) = x + (2y)\iota}}$

$\\ \implies { \bold{ [10 - 6\iota + 15\iota - 9 {\iota}^{2} ] (-4 - 7\iota) = x + (2y)\iota}}$

$\\ \implies { \bold{ [10 + 9\iota - 9 {\iota}^{2} ] (-4 - 7\iota) = x + (2y)\iota}}$

$\\ \rule {210}{1}$

☞ Let know about "iota" –

$\\ { \blue{\bold{ \: \: \: \: \: \: \: \: \: . \: \: \iota = \sqrt{ - 1} }}}$

$\\ { \blue{\bold{ \: \: \: \: \: \: \: \: \: . \: \: {\iota}^{2} = { - 1} }}}$

$\\ { \blue{\bold{ \: \: \: \: \: \: \: \: \: . \: \: {\iota}^{3} = - \iota }}}$

$\\ { \blue{\bold{ \: \: \: \: \: \: \: \: \: . \: \: {\iota}^{4} = 1}}}$

$\\ \rule {210}{1}$

• So that –

$\\ \implies { \bold{ [10 + 9\iota - 9 ( - 1) ] (-4 - 7\iota) = x + (2y)\iota}}$

$\\ \implies { \bold{ [10 + 9\iota + 9 ] (-4 - 7\iota) = x + (2y)\iota}}$

$\\ \implies { \bold{ [19 + 9\iota ] (-4 - 7\iota) = x + (2y)\iota}}$

$\\ \implies { \bold{ [19( - 4) + 19( - 7 \iota) + 9 \iota( - 4) + 9 \iota( - 7 \iota)] = x + (2y)\iota}}$

$\\ \implies { \bold{ [ - 76 - 133 \iota - 36 \iota - 63{ \iota}^{2} ] = x + (2y)\iota}}$

$\\ \implies { \bold{ [ - 76 - 133 \iota - 36 \iota - 63( - 1)] = x + (2y)\iota}}$

$\\ \implies { \bold{ (- 13 - 169 \iota ) = x + (2y)\iota}}$

• Now compare real and imaginary part –

$\\ \implies { \bold{ x = - 13 \: \: \: \: and \: \: \: \: 2y = - 169 \iota}}$

$\\ \implies { \bold{ x = - 13 \: \: \: \: and \: \: \: \: y = \frac {- 169 \iota}{2}}}$

$\rule {220}{4}$