The product of two integers is 112 . One number is four more then three times the other .
Which of the following equations could be used to find one of the numbers ?
A. 3x^2+4x=112
B. 4x^2+3=112
C. 4x^2+3x=112
D. 3x^2+4=112
The Correct Answer and Explanation is :
Let’s break down the problem:
We are given two key pieces of information:
- The product of two integers is 112.
- One of the integers is four more than three times the other.
Let the two integers be (x) and (y). According to the problem:
- (y = 3x + 4) (one number is four more than three times the other),
- The product of (x) and (y) is 112, so (x \cdot y = 112).
Now, substitute (y = 3x + 4) into the equation for the product of the two numbers:
[
x \cdot (3x + 4) = 112
]
Expanding the left side:
[
x(3x) + x(4) = 112
]
[
3x^2 + 4x = 112
]
This is a quadratic equation, which matches option A:
[
3x^2 + 4x = 112
]
Explanation:
- We were given that the product of the two numbers equals 112 and that one number is four more than three times the other. This led us to set up a relationship between (x) and (y).
- By substituting this relationship (y = 3x + 4) into the equation for their product, we derived a quadratic equation in terms of (x).
- The correct equation that could be used to find one of the numbers is (3x^2 + 4x = 112), which corresponds to option A.
Verifying the other options:
- Option B: (4x^2 + 3 = 112) — This does not match the problem, as there is no (3x) term and the numbers involved do not fit the scenario.
- Option C: (4x^2 + 3x = 112) — This equation does not fit the structure derived from the problem.
- Option D: (3x^2 + 4 = 112) — This is also incorrect because it lacks the (x) term needed to represent the relationship between the two integers.
Thus, the correct answer is A.