- Translate the problem requirements: We need to find the two solutions (roots) of the quadratic equation 2x² + 5x = 12, then calculate how much the larger root exceeds the smaller root (find the difference between them).
- Rearrange into standard quadratic form: Move all terms to one side to get the equation in the form ax² + bx + c = 0 so we can apply the quadratic formula.
- Apply the quadratic formula to find both roots: Use the formula x = (-b ± √(b² - 4ac))/2a to find the two solutions, where the ± gives us both the larger and smaller root.
- 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
Let's start by understanding what we're being asked to find. We have a quadratic equation: 2x² + 5x = 12, and we need to find its two solutions (called "roots"). Then we need to figure out by how much the larger solution exceeds the smaller one.
Think of it this way: if we had two numbers, say 8 and 3, then the larger number (8) exceeds the smaller number (3) by 8 - 3 = 5. We're doing the same thing here, but first we need to find those two numbers by solving the equation.
Process Skill: TRANSLATE - Converting the problem language "by how much does the larger root exceed the smaller root" into the mathematical operation of finding the difference between two roots
2. Rearrange into standard quadratic form
Right now our equation is 2x² + 5x = 12, but to use the standard method for solving quadratic equations, we need all terms on one side.
Let's move the 12 to the left side by subtracting 12 from both sides:
2x² + 5x - 12 = 0
Now we have our equation in the standard form ax² + bx + c = 0, where:
3. Apply the quadratic formula to find both roots
The quadratic formula tells us that for any equation ax² + bx + c = 0, the solutions are:
x = (-b ± √(b² - 4ac))/2a
Let's substitute our values a = 2, b = 5, and c = -12:
First, let's calculate the discriminant (the part under the square root):
b² - 4ac = 5² - 4(2)(-12) = 25 - (-96) = 25 + 96 = 121
Since √121 = 11, our formula becomes:
x = (-5 ± 11)/4
This gives us two solutions:
- Larger root: x = (-5 + 11)/4 = 6/4 = 3/2
- Smaller root: x = (-5 - 11)/4 = -16/4 = -4
Let's verify these work in our original equation 2x² + 5x = 12:
- For x = 3/2: 2(3/2)² + 5(3/2) = 2(9/4) + 15/2 = 9/2 + 15/2 = 24/2 = 12 ✓
- For x = -4: 2(-4)² + 5(-4) = 2(16) - 20 = 32 - 20 = 12 ✓
4. Calculate the difference between roots
Now we find by how much the larger root exceeds the smaller root:
Difference = Larger root - Smaller root
Difference = 3/2 - (-4) = 3/2 + 4 = 3/2 + 8/2 = 11/2
Final Answer
The larger root exceeds the smaller root by 11/2.
Looking at our answer choices, this matches option (E) 11/2.
Therefore, the answer is (E) 11/2.
Common Faltering Points
Errors while devising the approach
- Misunderstanding what "exceed" means: Students may confuse "by how much does the larger root exceed the smaller root" with finding the actual values of the roots themselves, or they might think they need to find the ratio between the roots rather than the difference. The key insight is that "exceed by" always means subtraction: larger value minus smaller value.
- Attempting to avoid the quadratic formula: Some students try to factor the equation 2x² + 5x - 12 = 0 instead of using the quadratic formula. While factoring can work, it's often more time-consuming for equations with non-integer coefficients, and students may waste valuable time or make errors trying to find factors that don't exist in simple integer form.
- Using shortcuts without understanding the full problem: Students might remember that for any quadratic ax² + bx + c = 0, there's a relationship between the roots, but they may try to apply Vieta's formulas (sum and product of roots) incorrectly without actually solving for the individual roots first.
Errors while executing the approach
- Sign errors when moving terms to standard form: When converting 2x² + 5x = 12 to 2x² + 5x - 12 = 0, students often make sign errors, particularly with the constant term. They might write 2x² + 5x + 12 = 0 instead, which completely changes the discriminant and leads to incorrect roots.
- Arithmetic mistakes in the discriminant calculation: The discriminant b² - 4ac = 5² - 4(2)(-12) involves multiple signs and operations. Students frequently make errors like: forgetting that 4(2)(-12) = -96, so b² - 4ac becomes 25 - (-96) = 25 + 96, not 25 - 96. This error leads to a negative discriminant and complex roots.
- Fraction arithmetic errors in the final calculation: When calculating the difference 3/2 - (-4), students may struggle with the double negative or the fraction arithmetic. Common errors include: writing 3/2 - (-4) as 3/2 - 4 instead of 3/2 + 4, or incorrectly adding 3/2 + 4 by not converting 4 to the common denominator 8/2.
Errors while selecting the answer
- Selecting one of the individual roots instead of the difference: After finding the roots 3/2 and -4, students might select 3/2 (which equals answer choice A) because they lose track of what the question is actually asking for. They solve correctly but forget the final step of finding the difference between the roots.
- Calculating the wrong difference: Students may calculate smaller root minus larger root instead of larger root minus smaller root, giving them -4 - 3/2 = -11/2. Since this is negative and doesn't match any answer choice, they might then just select 11/2 without understanding why their approach gave a negative result.
