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Answer :
To find which expression is equal to [tex]\((3x - 5)(2x - 7)\)[/tex], let’s go through the steps to expand and simplify the expression.
1. Distribute each term: Start by distributing each term in the first parentheses by each term in the second parentheses.
[tex]\[
(3x - 5)(2x - 7)
\][/tex]
2. Multiply the terms: Use the distributive property (also known as FOIL - First, Outer, Inner, Last) to expand the expression.
- First terms: [tex]\(3x \cdot 2x = 6x^2\)[/tex]
- Outer terms: [tex]\(3x \cdot -7 = -21x\)[/tex]
- Inner terms: [tex]\(-5 \cdot 2x = -10x\)[/tex]
- Last terms: [tex]\(-5 \cdot -7 = 35\)[/tex]
3. Combine like terms: Add the results of these multiplications together.
[tex]\[
6x^2 + (-21x) + (-10x) + 35
\][/tex]
[tex]\[
6x^2 - 21x - 10x + 35
\][/tex]
[tex]\[
6x^2 - 31x + 35
\][/tex]
4. Compare with the options: The expanded expression we obtained is [tex]\(6x^2 - 31x + 35\)[/tex]. Now we compare this with the given options:
- [tex]\(6x^2 + 31x - 35\)[/tex]
- [tex]\(6x^2 - 31x - 12\)[/tex]
- [tex]\(5x^2 - 21x + 12\)[/tex]
- [tex]\(6x^2 - 31x + 35\)[/tex]
The expression [tex]\(6x^2 - 31x + 35\)[/tex] matches exactly with the last option provided.
Therefore, the expression [tex]\((3x - 5)(2x - 7)\)[/tex] is equal to:
[tex]\[ \boxed{6x^2 - 31x + 35} \][/tex]
1. Distribute each term: Start by distributing each term in the first parentheses by each term in the second parentheses.
[tex]\[
(3x - 5)(2x - 7)
\][/tex]
2. Multiply the terms: Use the distributive property (also known as FOIL - First, Outer, Inner, Last) to expand the expression.
- First terms: [tex]\(3x \cdot 2x = 6x^2\)[/tex]
- Outer terms: [tex]\(3x \cdot -7 = -21x\)[/tex]
- Inner terms: [tex]\(-5 \cdot 2x = -10x\)[/tex]
- Last terms: [tex]\(-5 \cdot -7 = 35\)[/tex]
3. Combine like terms: Add the results of these multiplications together.
[tex]\[
6x^2 + (-21x) + (-10x) + 35
\][/tex]
[tex]\[
6x^2 - 21x - 10x + 35
\][/tex]
[tex]\[
6x^2 - 31x + 35
\][/tex]
4. Compare with the options: The expanded expression we obtained is [tex]\(6x^2 - 31x + 35\)[/tex]. Now we compare this with the given options:
- [tex]\(6x^2 + 31x - 35\)[/tex]
- [tex]\(6x^2 - 31x - 12\)[/tex]
- [tex]\(5x^2 - 21x + 12\)[/tex]
- [tex]\(6x^2 - 31x + 35\)[/tex]
The expression [tex]\(6x^2 - 31x + 35\)[/tex] matches exactly with the last option provided.
Therefore, the expression [tex]\((3x - 5)(2x - 7)\)[/tex] is equal to:
[tex]\[ \boxed{6x^2 - 31x + 35} \][/tex]
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