Identify the complex number and its conjugate.

Question

ScanSolve AI Tutor · Accepted Answer

To express the complex number in Cartesian form, we multiply the numerator and the denominator by the conjugate of the denominator.

Identify the complex number and its conjugate.
The given complex number is $\frac{2+5i}{-2-3i}$ .
The denominator is $-2-3i$ . Its conjugate is $-2+3i$ .
Multiply the numerator and denominator by the conjugate.
$\frac{2+5i}{-2-3i} \times \frac{-2+3i}{-2+3i}$
Expand the numerator and the denominator.
Numerator:
$(2+5i)(-2+3i) = 2(-2) + 2(3i) + 5i(-2) + 5i(3i)$ $= -4 + 6i - 10i + 15i^2$
Since $i^2 = -1$ :
$= -4 - 4i + 15(-1)$ $= -4 - 4i - 15 = -19 - 4i$
Denominator:
$(-2-3i)(-2+3i) = (-2)^2 - (3i)^2$ $= 4 - 9i^2$
Since $i^2 = -1$ :
$= 4 - 9(-1)$ $= 4 + 9 = 13$
Combine the simplified numerator and denominator.
$\frac{-19 - 4i}{13}$
Express in Cartesian form $a+bi$ .
$-\frac{19}{13} - \frac{4}{13}i$

Answer