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Answer :
Sure, let's break down the problem step-by-step to find the correct equation:
1. Understand the problem:
- We have two integers whose product is 112.
- One number is four more than three times the other.
2. Define the variables:
- Let [tex]\( x \)[/tex] be one of the integers.
- The other integer [tex]\( y \)[/tex] is expressed as "four more than three times the first number," which can be written as [tex]\( y = 3x + 4 \)[/tex].
3. Formulate the product equation:
- The product of these two integers is given by the equation [tex]\( x \cdot y = 112 \)[/tex].
4. Substitute [tex]\( y \)[/tex] in the product equation:
- Replace [tex]\( y \)[/tex] with [tex]\( 3x + 4 \)[/tex] in the equation:
[tex]\[ x \cdot (3x + 4) = 112 \][/tex]
5. Simplify the equation:
- Distribute [tex]\( x \)[/tex] to both terms inside the parentheses:
[tex]\[ 3x^2 + 4x = 112 \][/tex]
6. Rearrange the equation:
- Now, the equation looks like this:
[tex]\[ 3x^2 + 4x - 112 = 0 \][/tex]
Now, let's compare this with the given options:
- A. [tex]\( 3x^2 + 4x = 112 \)[/tex] – This matches our rearranged equation before moving the 112 to the left side, but since we simplified it without any alteration, it still represents the correct form before zeroing out the constant term.
- B. [tex]\( 3x^2 + 4 = 112 \)[/tex] – Incorrect, because there is no [tex]\( x \)[/tex] term on the left side of the equation.
- C. [tex]\( 4x^2 + 3x = 112 \)[/tex] – Incorrect, the coefficients for [tex]\( x \)[/tex] and [tex]\( x^2 \)[/tex] don’t match our equation.
- D. [tex]\( 4x^2 + 3 = 112 \)[/tex] – Incorrect, it’s similar to B with incorrect coefficients and missing the [tex]\( x \)[/tex] term.
Based on our calculations, the correct choice is:
[tex]\[ \boxed{A. \, 3x^2 + 4x = 112} \][/tex]
1. Understand the problem:
- We have two integers whose product is 112.
- One number is four more than three times the other.
2. Define the variables:
- Let [tex]\( x \)[/tex] be one of the integers.
- The other integer [tex]\( y \)[/tex] is expressed as "four more than three times the first number," which can be written as [tex]\( y = 3x + 4 \)[/tex].
3. Formulate the product equation:
- The product of these two integers is given by the equation [tex]\( x \cdot y = 112 \)[/tex].
4. Substitute [tex]\( y \)[/tex] in the product equation:
- Replace [tex]\( y \)[/tex] with [tex]\( 3x + 4 \)[/tex] in the equation:
[tex]\[ x \cdot (3x + 4) = 112 \][/tex]
5. Simplify the equation:
- Distribute [tex]\( x \)[/tex] to both terms inside the parentheses:
[tex]\[ 3x^2 + 4x = 112 \][/tex]
6. Rearrange the equation:
- Now, the equation looks like this:
[tex]\[ 3x^2 + 4x - 112 = 0 \][/tex]
Now, let's compare this with the given options:
- A. [tex]\( 3x^2 + 4x = 112 \)[/tex] – This matches our rearranged equation before moving the 112 to the left side, but since we simplified it without any alteration, it still represents the correct form before zeroing out the constant term.
- B. [tex]\( 3x^2 + 4 = 112 \)[/tex] – Incorrect, because there is no [tex]\( x \)[/tex] term on the left side of the equation.
- C. [tex]\( 4x^2 + 3x = 112 \)[/tex] – Incorrect, the coefficients for [tex]\( x \)[/tex] and [tex]\( x^2 \)[/tex] don’t match our equation.
- D. [tex]\( 4x^2 + 3 = 112 \)[/tex] – Incorrect, it’s similar to B with incorrect coefficients and missing the [tex]\( x \)[/tex] term.
Based on our calculations, the correct choice is:
[tex]\[ \boxed{A. \, 3x^2 + 4x = 112} \][/tex]
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