Math Problem Statement

Perform the indicated operations and write the complex number in the standard form a + bi. 2i + (6 - i)/(2 + 5i) = ___ + ___i.

Solution

The problem in the image asks to perform the indicated operations and write the complex number in standard form a+bia + bi, where aa is the real part and bb is the imaginary part.

The equation given is:

2i+6−i2+5i=_+_i2i + \frac{6 - i}{2 + 5i} = \_ + \_i

Step 1: Simplify the fraction 6−i2+5i\frac{6 - i}{2 + 5i}

We can multiply both the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator. The conjugate of 2+5i2 + 5i is 2−5i2 - 5i.

6−i2+5i×2−5i2−5i=(6−i)(2−5i)(2+5i)(2−5i)\frac{6 - i}{2 + 5i} \times \frac{2 - 5i}{2 - 5i} = \frac{(6 - i)(2 - 5i)}{(2 + 5i)(2 - 5i)}

Step 2: Multiply out both the numerator and denominator

Numerator:

= 12 - 30i - 2i + 5i^2$$ Since $$i^2 = -1$$, we replace $$5i^2$$ with $$-5$$: $$12 - 30i - 2i - 5 = 7 - 32i$$ #### Denominator: $$(2 + 5i)(2 - 5i) = 2^2 - (5i)^2 = 4 - 25i^2 = 4 + 25 = 29$$ ### Step 3: Simplify the fraction $$\frac{7 - 32i}{29} = \frac{7}{29} - \frac{32}{29}i$$ ### Step 4: Add this result to $$2i$$ Now add $$2i$$ to the result: $$2i + \left(\frac{7}{29} - \frac{32}{29}i\right) = \frac{7}{29} + \left(2 - \frac{32}{29}\right)i$$ Simplifying the imaginary part: $$2 - \frac{32}{29} = \frac{58}{29} - \frac{32}{29} = \frac{26}{29}$$ ### Final result: $$\frac{7}{29} + \frac{26}{29}i$$ Thus, the complex number in standard form is: $$\boxed{\frac{7}{29} + \frac{26}{29}i}$$ Would you like further clarification on any step, or have questions? ### Related questions: 1. How do you find the conjugate of a complex number? 2. What is the significance of multiplying by the conjugate in complex division? 3. Why does $$i^2 = -1$$ in complex numbers? 4. How would the result change if the real parts were different? 5. How do you interpret the real and imaginary parts geometrically? ### Tip: When simplifying complex fractions, always multiply by the conjugate to eliminate imaginary terms from the denominator.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Division of Complex Numbers
Standard Form of a Complex Number

Formulas

Standard form of complex numbers: a + bi
Multiplying by the conjugate: (a + bi)(a - bi) = a^2 + b^2

Theorems

Conjugate multiplication to eliminate imaginary parts from the denominator

Suitable Grade Level

Grades 9-12

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