Let's address both parts of your query step by step.

1. Identifying the Prompt that Uses Linear Algebra and Fourier Analysis:

Linear algebra often involves operations with matrices or systems of equations, while Fourier analysis deals with transforming data, such as signals, from the time domain to the frequency domain.

Breakdown of the Prompts:

• (a) "Calculate the number of possible outcomes for a series of 10 coin flips, and then estimate the probability of getting exactly 6 heads":

  • This is a combinatorics problem involving probability, not linear algebra or Fourier analysis.

• (b) "Solve the following system, and then interpret the solution: 2x + 3y = 5, 4x - y = 6":

  • This is a system of linear equations, which involves linear algebra, but it does not involve Fourier analysis.

• (c) "Analyze a speech signal in Praat using a matrix of sound wave data. Manipulate the matrix and extract key frequency components":

  • Analyzing sound data involves Fourier analysis (breaking down the sound signal into its frequency components), and working with a matrix of data is an aspect of linear algebra.

• (d) "Decompose a complex audio signal into its frequency components, and then estimate the likelihood of certain frequency patterns occurring in the signal":

  • This involves Fourier analysis but does not explicitly involve linear algebra.

Correct Answer:

(c) is the prompt that involves both linear algebra (manipulating a matrix of sound wave data) and Fourier analysis (extracting key frequency components).

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2. Step-by-Step Guide to Solve the Quadratic Equation:

The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Given values:

• $a = 123$
• $b = 1.5$
• $c = 2$

Step 1: Calculate the Discriminant

The discriminant is the part under the square root:

$\Delta = b^2 - 4ac$

Substitute the values:

$\Delta = (1.5)^2 - 4(123)(2) = 2.25 - 984 = -981.75$

Step 2: Check the Discriminant

Since the discriminant is negative ($\Delta < 0$), the solution will involve complex numbers.

Step 3: Compute the Complex Roots

The formula for the roots when the discriminant is negative becomes:

$x = \frac{-b \pm i\sqrt{|\Delta|}}{2a}$

Where $i$ represents the imaginary unit ($i = \sqrt{-1}$).

$x = \frac{-1.5 \pm i\sqrt{981.75}}{2(123)}$

Now, calculate $\sqrt{981.75}$:

$\sqrt{981.75} \approx 31.34$

Step 4: Final Expression for the Roots

Substitute the values into the equation:

$x = \frac{-1.5 \pm 31.34i}{246}$

Breaking it into the two possible solutions:

$x_1 = \frac{-1.5}{246} + \frac{31.34i}{246} \approx -0.0061 + 0.1274i$

$x_2 = \frac{-1.5}{246} - \frac{31.34i}{246} \approx -0.0061 - 0.1274i$

Thus, the two complex solutions are:

$x_1 \approx -0.0061 + 0.1274i$

$x_2 \approx -0.0061 - 0.1274i$

Questions to Deepen Understanding:

1. How can Fourier analysis be applied to real-world signal processing tasks?
2. What properties of matrices are crucial in analyzing large data sets, like sound wave matrices?
3. How does the discriminant determine the nature of the roots in a quadratic equation?
4. Can linear algebra be used in image processing? If so, how?
5. What are some practical applications of solving systems of linear equations?

Tip: When working with quadratic equations, always check the discriminant first to understand if the roots are real or complex.