Study Guide: Complex Numbers and Quadratic Equations in Precalculus

Q1. Write the expression $\frac{2-5i}{1-6i}$ in standard form $a+bi$.

Background

Topic: Complex Numbers – Division Using Conjugates

This question tests your ability to express a complex fraction in standard form by multiplying the numerator and denominator by the conjugate of the denominator.

Key Terms and Formulas:

• Complex Number: $a + bi$, where $a$ and $b$ are real numbers.

• Conjugate: The conjugate of $a + bi$ is $a - bi$.

• Standard Form: $a + bi$

• Key Property: $i^2 = -1$

Step-by-Step Guidance

1. Identify the conjugate of the denominator $1 - 6i$. The conjugate is $1 + 6i$.

2. Multiply both the numerator and denominator by the conjugate of the denominator to eliminate $i$ from the denominator: $\frac{2-5i}{1-6i} \times \frac{1+6i}{1+6i}$

3. Expand the numerator using the distributive property (FOIL): $(2-5i)(1+6i)$

4. Expand the denominator using the difference of squares: $(1-6i)(1+6i)$

5. Combine like terms and use $i^2 = -1$ to simplify both the numerator and denominator.

Try solving on your own before revealing the answer!

Q2. Choose the sentence that describes the character of the solutions to the quadratic equation.

Background

Topic: Quadratic Equations – Nature of Solutions

This question tests your understanding of how the discriminant ($b^2 - 4ac$) determines the type of solutions for a quadratic equation.

Key Terms and Formulas:

• Quadratic Equation: $ax^2 + bx + c = 0$

• Discriminant: $D = b^2 - 4ac$

• If $D > 0$: Two unequal real solutions

• If $D = 0$: One repeated real solution

• If $D < 0$: Two complex solutions (conjugates)

Step-by-Step Guidance

1. Identify the coefficients $a$, $b$, and $c$ from the quadratic equation (if given).

2. Calculate the discriminant $D = b^2 - 4ac$.

3. Compare the value of $D$ to zero to determine the nature of the solutions.

4. Match your result to the correct description:

   • Two complex solutions

   • Two unequal real solutions

   • One repeated real solution

Try solving on your own before revealing the answer!