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 (more negative, so smaller absolute value) equals the "greater" number (less negative, so larger absolute value) times 2, minus 4. We need to find the greater number.
2. Set up variables for the two negative numbers: Let the greater number be x (closer to zero) and express the lesser number in terms of x using the given relationship.
3. Create and solve the product equation: Use the fact that their product equals 160 to form a quadratic equation in terms of x.
4. Verify the solution makes sense: Check that our answer produces two negative numbers with the correct relationship and product.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what the problem is telling us in plain English:

We have two negative numbers. When we multiply them together, we get 160. Since both numbers are negative, their product is positive, which makes sense.

Now, when we talk about the "greater" and "lesser" of two negative numbers, we need to be careful. The "greater" negative number is the one closer to zero (like -5 is greater than -10). The "lesser" negative number is the one further from zero (more negative).

The problem tells us: "the lesser of the two numbers is 4 less than twice the greater."

So if the greater number is, say, \(-8\), then twice that would be \(2 \times (-8) = -16\), and 4 less than that would be \(-16 - 4 = -20\). So the lesser number would be \(-20\).

We need to find which negative number is the greater one.

Process Skill: TRANSLATE - Converting the relationship "4 less than twice the greater" into mathematical terms

2. Set up variables for the two negative numbers

Now let's use algebra to represent what we know:

Let \(\mathrm{x} = \text{the greater number}\) (remember, this is negative but closer to zero)

Since the lesser number is "4 less than twice the greater":

Lesser number = \(2\mathrm{x} - 4\)

Let's check this makes sense: if \(\mathrm{x} = -8\), then the lesser number = \(2(-8) - 4 = -16 - 4 = -20\).

Since \(-8 > -20\), this relationship works correctly.

3. Create and solve the product equation

We know that the product of these two numbers equals 160:

(Greater number) × (Lesser number) = 160

\(\mathrm{x} \times (2\mathrm{x} - 4) = 160\)

Expanding the left side:

\(2\mathrm{x}^2 - 4\mathrm{x} = 160\)

Moving everything to one side:

\(2\mathrm{x}^2 - 4\mathrm{x} - 160 = 0\)

We can simplify by dividing everything by 2:

\(\mathrm{x}^2 - 2\mathrm{x} - 80 = 0\)

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 \(+8\), because \((-10) \times (+8) = -80\) and \((-10) + (+8) = -2\).

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

This gives us \(\mathrm{x} = 10\) or \(\mathrm{x} = -8\).

Since the problem states both numbers are negative, x must be \(-8\).

4. Verify the solution makes sense

Let's check our answer:

If the greater number is \(-8\), then:

Lesser number = \(2(-8) - 4 = -16 - 4 = -20\)

Checking our conditions:

• Both numbers are negative: ✓ (\(-8\) and \(-20\) are both negative)

• \(-8 > -20\), so \(-8\) is indeed the greater: ✓

• Their product: \((-8) \times (-20) = 160\): ✓

• The lesser is 4 less than twice the greater: \(2(-8) - 4 = -20\): ✓

All conditions are satisfied!

Final Answer

The greater number is \(-8\).

Looking at our answer choices, this corresponds to choice D) \(-8\).

Common Faltering Points

Errors while devising the approach

1. Misunderstanding "greater" and "lesser" for negative numbers: Students often confuse which negative number is "greater." They might think \(-20\) is greater than \(-8\) because \(20 > 8\), forgetting that for negative numbers, the one closer to zero is actually greater. This leads to setting up the wrong variable relationships.

2. Incorrectly translating "4 less than twice the greater": Students may struggle with the phrase "4 less than" and might write it as \(4 - 2\mathrm{x}\) instead of \(2\mathrm{x} - 4\). The order matters significantly in this algebraic translation.

3. Forgetting that both numbers must be negative: Students might set up the equation correctly but fail to remember the constraint that both numbers are negative, which is crucial for determining which solution to accept from the quadratic equation.

Errors while executing the approach

1. Arithmetic errors when expanding and simplifying: When expanding \(\mathrm{x}(2\mathrm{x} - 4) = 160\), students commonly make sign errors or forget to move all terms to one side properly, leading to incorrect quadratic equations like \(2\mathrm{x}^2 - 4\mathrm{x} + 160 = 0\) instead of \(2\mathrm{x}^2 - 4\mathrm{x} - 160 = 0\).

2. Factoring mistakes in the quadratic: Students often struggle to find the correct factor pairs for \(\mathrm{x}^2 - 2\mathrm{x} - 80 = 0\). They might incorrectly identify factors that multiply to \(-80\) but don't add to \(-2\), leading to wrong solutions.

3. Accepting the wrong solution from the quadratic: After solving \((\mathrm{x} - 10)(\mathrm{x} + 8) = 0\) and getting \(\mathrm{x} = 10\) or \(\mathrm{x} = -8\), students might choose \(\mathrm{x} = 10\) without remembering that both numbers must be negative, thus selecting the mathematically valid but contextually incorrect solution.

Errors while selecting the answer

1. Selecting the lesser number instead of the greater: After correctly solving and finding that the two numbers are \(-8\) and \(-20\), students might select \(-20\) as their final answer, forgetting that the question specifically asks for the "greater" number.

2. Failing to verify the solution: Students might arrive at \(-8\) but not double-check that it satisfies all the given conditions, potentially missing calculation errors that would lead them to reconsider their answer choice.