Question

Solve the equation

z² +|z| = 0,
Where z is a complex number .​

Answer 1

Step-by-step explanation:

As we know that complex number is represented by

$z = a + ib \\ \\ |z| = \sqrt{ {a}^{2} + {b}^{2} }$

now put these values to the equation

${z}^{2} + |z| = 0 \\ \\ {(a + ib)}^{2} + \sqrt{ {a}^{2} + {b}^{2} } = 0 \\ \\ {a}^{2} + 2iab - {b}^{2} + \sqrt{ {a}^{2} + {b}^{2} } = 0 + 0i \\ \\ {a}^{2} - {b}^{2} + \sqrt{ {a}^{2} + {b}^{2} } + 2iab = 0 + 0i \\ \\ compare \: real \: and \: imaginary \: terms \\ \\ {a}^{2} - {b}^{2} + \sqrt{ {a}^{2} + {b}^{2} } = 0 \\ \\ 2ab = 0$

Now we can assume value of a=0

${0}^{2} - {b}^{2} + \sqrt{ {0}^{2} + {b}^{2} } = 0 \\ \\ - {b}^{2} + b = 0 \\ \\ b( - b + 1) = 0 \\ \\ b = 0 \\ \\ b = 1 \\ \\ put \: the \: value \: of \:b \: in \:equations \: \\ \\ 2ab = 0 \\ \\ = > a = 0 \: when \: b \: = 0$

put b=0

${a}^{2} - {0}^{2} + \sqrt{ {a}^{2} + {0}^{2} } = 0 \\ \\ {a}^{2} + a = 0 \\ \\ a(a + 1) = 0 \\ \\ a = 0 \\ \\ a = - 1$

Hence Solution of given equation are

$(0 + 0i) \\ \\ (0 + i) \\ \\ (0 - i)$

Hope it helps you.