- Translate the problem requirements: We need to determine which statements must always be true when a quadratic equation \(\mathrm{x}^2 + \mathrm{ax} + \mathrm{b} = 0\) has two different positive roots. "Two different positive roots" means both solutions are greater than zero and not equal to each other.
- Apply Vieta's formulas to connect roots with coefficients: Use the fundamental relationships between the sum and product of roots and the coefficients a and b to establish necessary conditions.
- Analyze the sign implications of positive roots: Since both roots are positive, determine what this tells us about the signs of their sum and product, which directly relate to the coefficients.
- Evaluate each statement systematically: Test each given condition (I, II, III) against our findings to determine which must always be true for any quadratic with two different positive roots.
Execution of Strategic Approach
1. Translate the problem requirements
Let's start by understanding what we're looking for. We have a quadratic equation \(\mathrm{x}^2 + \mathrm{ax} + \mathrm{b} = 0\) that has "two different positive roots." This means:
- Both solutions are greater than zero (positive)
- The two solutions are not the same number (different)
For example, if our roots were 2 and 5, both are positive and different. If our roots were -1 and 3, one is negative so this wouldn't qualify. If our roots were 4 and 4, they're positive but not different.
We need to figure out which of the three statements (I, II, III) about the coefficients a and b must ALWAYS be true whenever we have this situation.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical conditions
2. Apply Vieta's formulas to connect roots with coefficients
Here's where we connect the dots between what we know about the roots and what that tells us about the coefficients a and b.
Think about it this way: if I told you the two roots of a quadratic, you could work backwards to find the original equation. There are specific relationships that connect roots to coefficients.
Let's call our two positive roots \(\mathrm{r}_1\) and \(\mathrm{r}_2\). The key relationships are:
- Sum of roots: \(\mathrm{r}_1 + \mathrm{r}_2 = -\mathrm{a}\)
- Product of roots: \(\mathrm{r}_1 \times \mathrm{r}_2 = \mathrm{b}\)
These relationships come from expanding \((\mathrm{x} - \mathrm{r}_1)(\mathrm{x} - \mathrm{r}_2) = \mathrm{x}^2 - (\mathrm{r}_1 + \mathrm{r}_2)\mathrm{x} + \mathrm{r}_1\mathrm{r}_2\) and comparing to \(\mathrm{x}^2 + \mathrm{ax} + \mathrm{b}\).
3. Analyze the sign implications of positive roots
Now let's think about what happens when both roots are positive numbers.
Since \(\mathrm{r}_1 > 0\) and \(\mathrm{r}_2 > 0\):
- Sum of roots: \(\mathrm{r}_1 + \mathrm{r}_2\) is definitely positive (positive + positive = positive)
- Product of roots: \(\mathrm{r}_1 \times \mathrm{r}_2\) is definitely positive (positive × positive = positive)
Using our relationships from step 2:
- Since \(\mathrm{r}_1 + \mathrm{r}_2 = -\mathrm{a}\), and \(\mathrm{r}_1 + \mathrm{r}_2 > 0\), we have \(-\mathrm{a} > 0\), which means \(\mathrm{a} < 0\)
- Since \(\mathrm{r}_1 \times \mathrm{r}_2 = \mathrm{b}\), and \(\mathrm{r}_1 \times \mathrm{r}_2 > 0\), we have \(\mathrm{b} > 0\)
Process Skill: INFER - Drawing non-obvious conclusions about the signs of coefficients
4. Evaluate each statement systematically
Now let's check each statement against what we discovered:
Statement I: \(\mathrm{a} < 0\)
From our analysis, we found that \(\mathrm{a} < 0\) must ALWAYS be true. ✓
Statement II: \(\mathrm{b} < 0\)
From our analysis, we found that \(\mathrm{b} > 0\) (b is positive). So \(\mathrm{b} < 0\) is NEVER true. ✗
Statement III: \(\mathrm{ab} < 0\)
We know \(\mathrm{a} < 0\) and \(\mathrm{b} > 0\). When we multiply a negative number by a positive number, we get a negative result. So \(\mathrm{ab} < 0\) is ALWAYS true. ✓
Therefore, statements I and III must be true, while statement II is false.
Process Skill: APPLY CONSTRAINTS - Using the conditions systematically to test each option
5. Final Answer
The statements that must be true are I and III only.
Looking at the answer choices:
- III only - No, both I and III are true
- I and II only - No, II is false
- I and III only - Yes, this matches our findings
- II and III only - No, II is false
- I, II, and III - No, II is false
Answer: C
Common Faltering Points
Errors while devising the approach
- Misinterpreting "two different positive roots": Students may focus only on the "positive" part and forget that the roots must also be "different." This leads them to miss checking the discriminant condition or not fully understanding what constraints this places on the coefficients.
- Not recognizing the need for Vieta's formulas: Students may try to solve this by attempting to factor specific examples or by trying to find actual numerical roots, rather than recognizing that Vieta's formulas directly connect the coefficients to properties of the roots without needing to solve for the roots themselves.
- Confusing the direction of logical reasoning: Students may think they need to check if each statement could be true in some cases, rather than understanding they need to prove which statements must ALWAYS be true whenever the given conditions are met.
Errors while executing the approach
- Sign errors when applying Vieta's formulas: Students frequently confuse that the sum of roots equals -a (negative coefficient), not +a. They may write \(\mathrm{r}_1 + \mathrm{r}_2 = \mathrm{a}\) instead of \(\mathrm{r}_1 + \mathrm{r}_2 = -\mathrm{a}\), leading to incorrect conclusions about the sign of coefficient a.
- Incorrect sign analysis: When determining that both roots are positive, students may incorrectly conclude the signs of the coefficients. For example, they might think that since the product of roots is positive, and this equals b, then b must be negative (confusing the relationship).
- Forgetting to check all implications: Students may correctly determine that \(\mathrm{a} < 0\) and \(\mathrm{b} > 0\), but then fail to evaluate what this means for statement III (\(\mathrm{ab} < 0\)), missing the compound relationship between the coefficients.
Errors while selecting the answer
- Selecting statements that are sometimes true rather than always true: Students may identify that certain statements can be true in specific examples, but fail to recognize that the question asks which must be true in ALL cases meeting the given conditions.
- Misreading the answer choice format: Students may correctly identify that statements I and III are true while II is false, but then select an answer choice like "I and II only" or "III only" due to careless reading of the options.
