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Complex Numbers

If z = 2 -3...

Question

If $z = 2 - 3 i$ , then show that $z^{2} - 4 z + 13 = 0$ . Hence find the value of $4 z^{3} - 3 z^{2} + 2 z + 170$ .

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Solution

Given,

$z = (2 - 3 i)$

$z^{2} - 4 z + 13$

$= {(2 - 3 i)}^{2} - 4 (2 - 3 i) + 13$

$= - 5 - 12 i - 8 + 12 i + 13$

$= - 12 i + 12 i - 5 - 8 + 13$

$= 0$

$4 z^{3} - 3 z^{2} + 2 z + 170$

$= 4 {(2 - 3 i)}^{3} - 3 {(2 - 3 i)}^{2} + 2 (2 - 3 i) + 170$

$= 4 (- 46 - 9 i) - 3 (- 5 - 12 i) + 2 (2 - 3 i) + 170$

$= - 184 - 36 i - 3 (- 5 - 12 i) + 2 (2 - 3 i) + 170$

$= - 184 - 36 i + 15 + 36 i + 4 - 6 i + 170$

$= 5 - 6 i$

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Similar questions

z = 2 - 3 i

, then show that,

z^{2} - 4 z + 13 = 0

z = 2 - 3 i

z^{2} - 4 z + 13 =

| z_{1} | = 2, | z_{2} | = 3, | z_{3} | = 4

| {2 z}_{1} + 3 z_{2} + {4 z}_{3} | = 4

then the absolute value of

{8 z}_{3} z_{2} + {27 z}_{3} z_{1} + {64 z}_{1} z_{2}

| z_{1} | = 2, | z_{2} | = 3, | z_{3} | = 4

| 2 z_{1} + 3 z_{2} + 4 z_{3} | = 0

, then absolute value of

8 z_{2} z_{2} + 27 z_{3} z_{1} + 64 z_{1} z_{2}

must be equal to

z = - 1 + 3 i

z^{2} + 2 z + 10 =

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