The product of two negative numbers is 160. If the lesser of the two numbers is 4 less than twice...

1. Translate the problem requirements: We have two negative numbers whose product is 160. The lesser number is 4 less than twice the greater number. We need to find the greater number.
2. Set up the relationship between the numbers: Define variables for the greater and lesser numbers and express their relationship algebraically.
3. Apply the product constraint: Use the fact that their product equals 160 to create an equation we can solve.
4. Solve the quadratic equation: Expand and rearrange to find the possible values for the greater number.
5. Verify the solution meets all constraints: Check that both numbers are negative and satisfy the original relationships.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in plain English: We have two negative numbers that when multiplied together give us 160. One of these numbers is smaller (lesser) than the other, and this smaller number has a specific relationship to the larger one - it's exactly 4 less than twice the larger number.

Since we're looking for the greater number, let's think about what makes sense. Both numbers are negative, so when we multiply two negative numbers, we get a positive result (160). The phrase "lesser of the two numbers" means the number that's more negative (further from zero on the number line).

Process Skill: TRANSLATE - Converting the word relationships into mathematical understanding

2. Set up the relationship between the numbers

Now let's assign variables to make this concrete. Let's call the greater number g (remember, it's negative, so it's closer to zero). The lesser number we'll call l (it's more negative, further from zero).

The relationship tells us: "the lesser number is 4 less than twice the greater number"
In everyday terms: If I take the greater number, double it, then subtract 4, I get the lesser number.

So: \(\mathrm{l = 2g - 4}\)

Let's verify this makes sense: If \(\mathrm{g = -8}\), then \(\mathrm{l = 2(-8) - 4 = -16 - 4 = -20}\). Since -20 is indeed less than -8, this relationship works correctly.

3. Apply the product constraint

We know that when we multiply our two numbers together, we get 160:
\(\mathrm{g \times l = 160}\)

Now we can substitute our relationship from step 2:
\(\mathrm{g \times (2g - 4) = 160}\)

This gives us one equation with one unknown, which we can solve.

4. Solve the quadratic equation

Let's expand our equation step by step:
\(\mathrm{g \times (2g - 4) = 160}\)
\(\mathrm{2g^2 - 4g = 160}\)

Rearranging to standard form:
\(\mathrm{2g^2 - 4g - 160 = 0}\)

We can simplify by dividing everything by 2:
\(\mathrm{g^2 - 2g - 80 = 0}\)

Now we need to factor this quadratic. We're looking for two numbers that multiply to -80 and add to -2.
Those numbers are -10 and +8: (-10) × (+8) = -80 and (-10) + (+8) = -2

So: \(\mathrm{(g - 10)(g + 8) = 0}\)

This gives us: \(\mathrm{g = 10}\) or \(\mathrm{g = -8}\)

Process Skill: APPLY CONSTRAINTS - We must check which solution fits our problem constraints

5. Verify the solution meets all constraints

Since the problem states both numbers are negative, \(\mathrm{g = 10}\) doesn't work (it's positive).

So \(\mathrm{g = -8}\) is our greater number.

Let's verify: If \(\mathrm{g = -8}\), then \(\mathrm{l = 2(-8) - 4 = -16 - 4 = -20}\)

Checking our constraints:
✓ Both numbers are negative: -8 and -20
✓ -8 is greater than -20 (closer to zero)
✓ Product check: \(\mathrm{(-8) \times (-20) = 160}\) ✓
✓ Relationship check: \(\mathrm{-20 = 2(-8) - 4 = -16 - 4 = -20}\) ✓

4. Final Answer

The greater number is -8, which corresponds to answer choice D.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "lesser" and "greater" for negative numbers

Students often confuse which negative number is "greater." They might think -20 is greater than -8 because 20 > 8, forgetting that on the number line, -8 is actually greater than -20 (closer to zero). This leads to setting up the wrong variable assignments and relationships.

2. Incorrectly translating "4 less than twice the greater"

Students may set up the relationship as \(\mathrm{g = 2l - 4}\) instead of \(\mathrm{l = 2g - 4}\). They confuse which number the relationship is describing, leading to an entirely wrong equation setup.

3. Missing the constraint that both numbers must be negative

Students might overlook this critical constraint stated in the problem, which becomes crucial when choosing between the two solutions of the quadratic equation in later steps.

Errors while executing the approach

1. Sign errors when expanding the equation

When expanding \(\mathrm{g \times (2g - 4) = 160}\), students often make sign errors, particularly with the -4g term, leading to incorrect quadratic equations like \(\mathrm{2g^2 + 4g = 160}\) instead of \(\mathrm{2g^2 - 4g = 160}\).

2. Factoring errors in the quadratic equation

When factoring \(\mathrm{g^2 - 2g - 80 = 0}\), students may incorrectly identify the factor pairs. They need two numbers that multiply to -80 and add to -2, but might choose wrong combinations like -8 and +10, leading to incorrect factors.

3. Arithmetic mistakes in verification

Even with correct setup, students may make calculation errors when checking their answer, such as computing \(\mathrm{(-8) \times (-20)}\) incorrectly or making errors in \(\mathrm{l = 2(-8) - 4}\).

Errors while selecting the answer

1. Choosing the wrong solution from the quadratic

After correctly solving \(\mathrm{(g - 10)(g + 8) = 0}\) to get \(\mathrm{g = 10}\) or \(\mathrm{g = -8}\), students might select \(\mathrm{g = 10}\) as the answer, forgetting that the problem specifies both numbers must be negative.

2. Reporting the lesser number instead of the greater number

Students might correctly find both numbers (-8 and -20) but then report -20 as the answer, forgetting that the question specifically asks for the greater number.