QuestionExpress $\frac{2 i+3}{5 i+2}$ in the form of $a + b i$ .

Studdy Solution

STEP 1

Assumptions1. The given expression is $\frac{ i+3}{5 i+}$ . We are asked to express this in the form $a+bi$ , where $a$ and $b$ are real numbers.

STEP 2

We can multiply the numerator and denominator of the given expression by the conjugate of the denominator to eliminate the imaginary part from the denominator.
The conjugate of a complex number $a+bi$ is $a-bi$ . So, the conjugate of $5i+2$ is $5i-2$ .

STEP 3

Multiply the numerator and denominator by the conjugate of the denominator.
$\frac{2 i+3}{5 i+2} \times \frac{5i-2}{5i-2}$

STEP 4

Multiply the numerators together.
$(2i+3)(i-2) =10i^2 -4i +15i -6$

STEP 5

implify the numerator. Remember that $i^2 = -1$ .
$10i^2 -4i +15i - =10(-1) -4i +15i - = -10 -4i +15i - = -16 +11i$

STEP 6

Multiply the denominators together.
$(5i+2)(5i-2) =25i^2 -4 =25(-1) -4 = -25 -4 = -29$

STEP 7

Now, divide the numerator by the denominator.
$\frac{-16 +11i}{-29} = \frac{-16}{-29} + \frac{11i}{-29} = \frac{16}{29} - \frac{11}{29}i$ So, $a = \frac{16}{29}$ and $b = -\frac{11}{29}$ .
The given expression in the form $a+bi$ is $\frac{16}{29} - \frac{11}{29}i$ .

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