Answer :
Let's solve the problem step by step.
We need to find an equation that could be used to find one of the numbers, given that the product of two integers is [tex]\(112\)[/tex], and one number is four more than three times the other.
Let's denote the two unknown integers as [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. We're given two conditions:
1. The product of the integers is 112.
2. One number is four more than three times the other.
#### Step 1: Express the Given Conditions
1. The product of the integers:
[tex]\[
x \cdot y = 112
\][/tex]
2. One number is four more than three times the other. Let's assume that [tex]\(x\)[/tex] is four more than three times [tex]\(y\)[/tex]:
[tex]\[
x = 3y + 4
\][/tex]
#### Step 2: Substitute the Second Condition into the First
We can substitute [tex]\(x\)[/tex] from the second condition into the first condition:
[tex]\[
(3y + 4) \cdot y = 112
\][/tex]
#### Step 3: Simplify the Equation
Now, let's simplify this equation:
[tex]\[
3y^2 + 4y = 112
\][/tex]
#### Step 4: Rearrange to Match the Standard Form of a Quadratic Equation
We need to put this equation in the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
3y^2 + 4y - 112 = 0
\][/tex]
#### Step 5: Identify the Correct Equation
The simplified equation [tex]\(3y^2 + 4y - 112 = 0\)[/tex] matches one of the given options with the correct structure. We can identify that the correct option is:
[tex]\[
\text{A. } 3x^2 + 4x = 112
\][/tex]
Therefore, the equation that could be used to find one of the numbers is:
[tex]\[
\boxed{3x^2 + 4x = 112}
\][/tex]
Option A is the correct answer.
We need to find an equation that could be used to find one of the numbers, given that the product of two integers is [tex]\(112\)[/tex], and one number is four more than three times the other.
Let's denote the two unknown integers as [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. We're given two conditions:
1. The product of the integers is 112.
2. One number is four more than three times the other.
#### Step 1: Express the Given Conditions
1. The product of the integers:
[tex]\[
x \cdot y = 112
\][/tex]
2. One number is four more than three times the other. Let's assume that [tex]\(x\)[/tex] is four more than three times [tex]\(y\)[/tex]:
[tex]\[
x = 3y + 4
\][/tex]
#### Step 2: Substitute the Second Condition into the First
We can substitute [tex]\(x\)[/tex] from the second condition into the first condition:
[tex]\[
(3y + 4) \cdot y = 112
\][/tex]
#### Step 3: Simplify the Equation
Now, let's simplify this equation:
[tex]\[
3y^2 + 4y = 112
\][/tex]
#### Step 4: Rearrange to Match the Standard Form of a Quadratic Equation
We need to put this equation in the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
3y^2 + 4y - 112 = 0
\][/tex]
#### Step 5: Identify the Correct Equation
The simplified equation [tex]\(3y^2 + 4y - 112 = 0\)[/tex] matches one of the given options with the correct structure. We can identify that the correct option is:
[tex]\[
\text{A. } 3x^2 + 4x = 112
\][/tex]
Therefore, the equation that could be used to find one of the numbers is:
[tex]\[
\boxed{3x^2 + 4x = 112}
\][/tex]
Option A is the correct answer.
