By how much does the larger root of the equation 2x^2 + 5x = 12 exceed the smaller root?

1. Translate the problem requirements: We need to find the two roots (solutions) of the equation \(2\mathrm{x}^2 + 5\mathrm{x} = 12\), identify which is larger and which is smaller, then calculate how much the larger exceeds the smaller by finding their difference.
2. Rearrange to standard quadratic form: Move all terms to one side to get the equation in the form \(\mathrm{a}\mathrm{x}^2 + \mathrm{b}\mathrm{x} + \mathrm{c} = 0\) so we can solve for the values of x.
3. Find both roots by factoring: Look for two numbers that multiply to give the constant term and add to give the middle coefficient, then use these to factor the quadratic completely.
4. Calculate the difference between roots: Subtract the smaller root from the larger root to find by how much the larger exceeds the smaller.

Execution of Strategic Approach

1. Translate the problem requirements

We need to understand exactly what we're looking for. The equation \(2\mathrm{x}^2 + 5\mathrm{x} = 12\) has two solutions (called roots). Think of it like this: there are two different values of x that make this equation true. One of these values will be larger than the other. The question asks us to find how much bigger the larger value is compared to the smaller value.

In everyday terms, if the larger root is 3 and the smaller root is -1, then the larger root exceeds the smaller by \(3 - (-1) = 4\).

Process Skill: TRANSLATE - Converting the mathematical language into clear understanding of what we need to find

2. Rearrange to standard quadratic form

Right now our equation is \(2\mathrm{x}^2 + 5\mathrm{x} = 12\). To solve this properly, we need all terms on one side with zero on the other side. This is like organizing everything neatly before we start working.

Starting with: \(2\mathrm{x}^2 + 5\mathrm{x} = 12\)
Subtract 12 from both sides: \(2\mathrm{x}^2 + 5\mathrm{x} - 12 = 0\)

Now we have our equation in standard form: \(2\mathrm{x}^2 + 5\mathrm{x} - 12 = 0\)
This means \(\mathrm{a} = 2\), \(\mathrm{b} = 5\), and \(\mathrm{c} = -12\)

3. Find both roots by factoring

Now we need to factor \(2\mathrm{x}^2 + 5\mathrm{x} - 12 = 0\). We're looking for two numbers that work together to create our equation.

For a quadratic like \(2\mathrm{x}^2 + 5\mathrm{x} - 12\), we need two numbers that:
- Multiply to give \((2)(-12) = -24\)
- Add to give 5

Let's think: what two numbers multiply to -24 and add to 5?
Let's try different pairs: 8 and -3 work! (\(8 \times (-3) = -24\), and \(8 + (-3) = 5\))

Now we rewrite the middle term:
\(2\mathrm{x}^2 + 5\mathrm{x} - 12 = 2\mathrm{x}^2 + 8\mathrm{x} - 3\mathrm{x} - 12\)

Group the terms:
\(= 2\mathrm{x}(\mathrm{x} + 4) - 3(\mathrm{x} + 4)\)
\(= (2\mathrm{x} - 3)(\mathrm{x} + 4)\)

So our equation becomes: \((2\mathrm{x} - 3)(\mathrm{x} + 4) = 0\)

For this to equal zero, either:
\(2\mathrm{x} - 3 = 0\) or \(\mathrm{x} + 4 = 0\)

From \(2\mathrm{x} - 3 = 0\): \(\mathrm{x} = \frac{3}{2}\)
From \(\mathrm{x} + 4 = 0\): \(\mathrm{x} = -4\)

Our two roots are \(\mathrm{x} = \frac{3}{2}\) and \(\mathrm{x} = -4\)

4. Calculate the difference between roots

Now we identify which root is larger and which is smaller:
- \(\frac{3}{2} = 1.5\)
- \(-4 = -4\)

Clearly, \(\frac{3}{2}\) is the larger root and \(-4\) is the smaller root.

The difference is: Larger root - Smaller root
\(= \frac{3}{2} - (-4)\)
\(= \frac{3}{2} + 4\)
\(= \frac{3}{2} + \frac{8}{2}\)
\(= \frac{11}{2}\)

Process Skill: MANIPULATE - Carefully handling the subtraction of negative numbers

Final Answer

The larger root exceeds the smaller root by \(\frac{11}{2}\).

Looking at our answer choices, this matches choice (E) \(\frac{11}{2}\).

Let's verify by checking our roots in the original equation:
For \(\mathrm{x} = \frac{3}{2}\): \(2\left(\frac{3}{2}\right)^2 + 5\left(\frac{3}{2}\right) = 2\left(\frac{9}{4}\right) + \frac{15}{2} = \frac{9}{2} + \frac{15}{2} = \frac{24}{2} = 12\) ✓
For \(\mathrm{x} = -4\): \(2(-4)^2 + 5(-4) = 2(16) - 20 = 32 - 20 = 12\) ✓

The answer is (E) \(\frac{11}{2}\).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what "exceed" means: Students might think they need to find the ratio of the larger root to the smaller root, or the percentage by which one exceeds the other, rather than the simple arithmetic difference (larger - smaller).

2. Not recognizing this as a standard quadratic equation problem: Some students might get overwhelmed by the wording and not realize they simply need to solve \(2\mathrm{x}^2 + 5\mathrm{x} = 12\) for both roots first before finding their difference.

3. Forgetting to rearrange to standard form: Students might attempt to solve \(2\mathrm{x}^2 + 5\mathrm{x} = 12\) directly without moving all terms to one side, leading them to use incorrect methods or formulas.

Errors while executing the approach

1. Factoring errors: When factoring \(2\mathrm{x}^2 + 5\mathrm{x} - 12\), students commonly make mistakes in finding the correct pair of numbers that multiply to \(\mathrm{ac} = -24\) and add to \(\mathrm{b} = 5\). They might choose incorrect pairs like 6 and -4, or make sign errors.

2. Arithmetic mistakes when solving for individual roots: Even with correct factoring, students often make errors when solving \(2\mathrm{x} - 3 = 0\) (forgetting to divide by 2) or when solving \(\mathrm{x} + 4 = 0\) (sign errors), leading to incorrect root values.

3. Errors in calculating the final difference: The subtraction \(\frac{3}{2} - (-4)\) is particularly error-prone because students often struggle with subtracting negative numbers, potentially calculating \(\frac{3}{2} - 4 = -\frac{5}{2}\) instead of \(\frac{3}{2} + 4 = \frac{11}{2}\).

Errors while selecting the answer

1. Calculating the difference in wrong order: Students might subtract the larger root from the smaller root (calculating \(-4 - \frac{3}{2} = -\frac{11}{2}\)) and then either select a negative answer if available, or take the absolute value and get confused about which answer choice matches.

2. Converting fractions incorrectly: When getting \(\frac{11}{2}\) as their answer, students might incorrectly convert this to a decimal (5.5) and then try to match it with answer choices, potentially misidentifying it as \(\frac{5}{2}\) or making other conversion errors.