1. Key Concepts and Formulas

To solve this problem, we'll utilize fundamental concepts from statistics and algebra:

• Mean ($\bar{x}$): The average of a set of $n$ observations $x_1, x_2, \dots, x_n$. $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$
• Variance ($\sigma^2$): A measure of how spread out the numbers are. It is the square of the standard deviation ($\sigma$). A commonly used computational formula for variance is: $\sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - (\bar{x})^2$
• Standard Deviation ($\sigma$): The positive square root of the variance, indicating the typical deviation from the mean. $\sigma = \sqrt{\sigma^2}$
• Quadratic Equation from Roots: If $a$ and $b$ are the roots of a quadratic equation, the equation can be expressed as: $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$ Here, the sum of roots is $(a+b)$ and the product of roots is $ab$.
• Algebraic Identity: A useful identity to relate the sum, sum of squares, and product of two numbers: $(a+b)^2 = a^2 + b^2 + 2ab$

2. Step-by-Step Solution

Our strategy involves using the given mean and standard deviation to find the sum $(a+b)$ and the product $(ab)$ of the unknown values $a$ and $b$. Once these are found, we can construct the quadratic equation.

Given Information:

• Data set: $3, 5, 7, a, b$
• Number of observations ($n$): $5$
• Mean ($\bar{x}$): $5$
• Standard deviation ($\sigma$): $2$

Step 1: Find the Sum of $a$ and $b$ using the Mean

• What we are doing and WHY: We use the definition of the mean to establish a relationship between the known data points and the sum of the unknowns $(a+b)$. This will give us the sum of the roots for our quadratic equation.
• Math: The formula for the mean is $\bar{x} = \frac{\sum x_i}{n}$. Substitute the given mean and data points: $5 = \frac{3 + 5 + 7 + a + b}{5}$
• Reasoning and Calculation: First, sum the known numerical values in the data set: $5 = \frac{15 + a + b}{5}$ Multiply both sides by $5$ to isolate the numerator: $5 \times 5 = 15 + a + b$ $25 = 15 + a + b$ Subtract $15$ from both sides to find the sum $(a+b)$: $a + b = 25 - 15$ $a + b = 10 \quad \text{(Equation 1)}$ This is the sum of the roots of the quadratic equation.

Step 2: Find the Sum of Squares ($a^2+b^2$) using Standard Deviation

• What we are doing and WHY: We use the standard deviation to calculate the variance. The variance formula involves the sum of squares of all observations, which will help us find $a^2+b^2$.
• Math: First, calculate the variance from the given standard deviation: $\sigma^2 = (\text{standard deviation})^2 = 2^2 = 4$ Next, calculate the sum of squares of all observations: $\sum x_i^2 = 3^2 + 5^2 + 7^2 + a^2 + b^2$ $\sum x_i^2 = 9 + 25 + 49 + a^2 + b^2$ $\sum x_i^2 = 83 + a^2 + b^2$ Now, substitute $\sigma^2$, $\sum x_i^2$, and $\bar{x}$ into the variance formula: $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$ $4 = \frac{83 + a^2 + b^2}{5} - (5)^2$
• Reasoning and Calculation: Simplify the $(\bar{x})^2$ term: $4 = \frac{83 + a^2 + b^2}{5} - 25$ Add $25$ to both sides to isolate the fraction term: $4 + 25 = \frac{83 + a^2 + b^2}{5}$ $29 = \frac{83 + a^2 + b^2}{5}$ Multiply both sides by $5$ to eliminate the denominator: $29 \times 5 = 83 + a^2 + b^2$ $145 = 83 + a^2 + b^2$ Subtract $83$ from both sides to find the sum of squares $(a^2+b^2)$: $a^2 + b^2 = 145 - 83$ $a^2 + b^2 = 62 \quad \text{(Equation 2)}$

Step 3: Find the Product of $a$ and $b$ using an Algebraic Identity

• What we are doing and WHY: We now have the sum $(a+b)$ from Step 1 and the sum of squares $(a^2+b^2)$ from Step 2. We use the algebraic identity $(a+b)^2 = a^2+b^2+2ab$ to find the product $ab$, which is the remaining component needed for the quadratic equation.
• Math: The algebraic identity is $(a+b)^2 = a^2 + b^2 + 2ab$. Substitute the values from Equation 1 ($a+b=10$) and Equation 2 ($a^2+b^2=62$): $(10)^2 = 62 + 2ab$
• Reasoning and Calculation: Simplify $(10)^2$: $100 = 62 + 2ab$ Subtract $62$ from both sides to isolate the $2ab$ term: $100 - 62 = 2ab$ $38 = 2ab$ Divide both sides by $2$ to find the product $ab$: $ab = \frac{38}{2}$ $ab = 19 \quad \text{(Equation 3)}$ This is the product of the roots of the quadratic equation.

Step 4: Form the Quadratic Equation

• What we are doing and WHY: With the sum of roots $(a+b)$ and the product of roots $(ab)$ determined, we can now directly write the quadratic equation.
• Math: The general form of a quadratic equation with roots $a$ and $b$ is $x^2 - (a+b)x + ab = 0$. Substitute the values from Equation 1 ($a+b=10$) and Equation 3 ($ab=19$): $x^2 - (10)x + 19 = 0$
• Final Equation: $x^2 - 10x + 19 = 0$

3. Common Mistakes & Tips

• Confusing $\sum x_i^2$ and $(\sum x_i)^2$: Remember that $\sum x_i^2$ means summing the squares of individual observations (e.g., $3^2+5^2+7^2+a^2+b^2$), while $(\sum x_i)^2$ means squaring the sum of all observations (e.g., $(3+5+7+a+b)^2$).
• Incorrect Variance Formula: Always use the correct formula for variance. The one used here, $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$, is very efficient.
• Algebraic Errors: Be careful with arithmetic operations, especially when isolating terms like $a+b$, $a^2+b^2$, and $ab$.
• Connecting Concepts: The trick to such problems is to understand how statistical measures (mean, standard deviation) provide algebraic information (sum of roots, sum of squares of roots), which can then be used to form a polynomial equation.

4. Summary

We systematically used the given mean and standard deviation to extract the necessary information about the unknowns $a$ and $b$. First, the mean allowed us to find the sum $(a+b)$. Then, the standard deviation (via variance) helped us determine the sum of squares $(a^2+b^2)$. Finally, an algebraic identity was used to link these two sums to find the product $(ab)$. With the sum and product of roots, the quadratic equation was directly formed. The derived equation is $x^2 - 10x + 19 = 0$.

5. Final Answer

The quadratic equation whose roots are $a$ and $b$ is $x^2 - 10x + 19 = 0$. This corresponds to option (D).

The final answer is $\boxed{x^2 - 10x + 19 = 0}$.