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Answer :
To find which expression is equal to [tex]\((3x - 5)(2x - 7)\)[/tex], we can expand the expression using the distributive property, also known as the FOIL method. FOIL stands for First, Outer, Inner, Last, which are the pairs of terms we multiply together.
Let's go through the steps:
1. First terms: Multiply the first terms in each binomial: [tex]\(3x \times 2x = 6x^2\)[/tex].
2. Outer terms: Multiply the outer terms of the binomials: [tex]\(3x \times (-7) = -21x\)[/tex].
3. Inner terms: Multiply the inner terms: [tex]\(-5 \times 2x = -10x\)[/tex].
4. Last terms: Multiply the last terms of each binomial: [tex]\(-5 \times (-7) = 35\)[/tex].
Now, combine all these results to form the expanded expression:
[tex]\[
6x^2 + (-21x) + (-10x) + 35
\][/tex]
Combine the like terms ([tex]\(-21x\)[/tex] and [tex]\(-10x\)[/tex]):
[tex]\[
6x^2 - 21x - 10x + 35 = 6x^2 - 31x + 35
\][/tex]
Thus, the expanded expression is [tex]\(6x^2 - 31x + 35\)[/tex], which matches one of the given choices:
- [tex]\(6x^2 + 31x - 35\)[/tex]
- [tex]\(6x^2 - 31x + 35\)[/tex]
- [tex]\(6x^2 - 31x - 12\)[/tex]
- [tex]\(5x^2 - 21x + 12\)[/tex]
Therefore, the expression [tex]\((3x-5)(2x-7)\)[/tex] is equal to [tex]\(6x^2 - 31x + 35\)[/tex].
Let's go through the steps:
1. First terms: Multiply the first terms in each binomial: [tex]\(3x \times 2x = 6x^2\)[/tex].
2. Outer terms: Multiply the outer terms of the binomials: [tex]\(3x \times (-7) = -21x\)[/tex].
3. Inner terms: Multiply the inner terms: [tex]\(-5 \times 2x = -10x\)[/tex].
4. Last terms: Multiply the last terms of each binomial: [tex]\(-5 \times (-7) = 35\)[/tex].
Now, combine all these results to form the expanded expression:
[tex]\[
6x^2 + (-21x) + (-10x) + 35
\][/tex]
Combine the like terms ([tex]\(-21x\)[/tex] and [tex]\(-10x\)[/tex]):
[tex]\[
6x^2 - 21x - 10x + 35 = 6x^2 - 31x + 35
\][/tex]
Thus, the expanded expression is [tex]\(6x^2 - 31x + 35\)[/tex], which matches one of the given choices:
- [tex]\(6x^2 + 31x - 35\)[/tex]
- [tex]\(6x^2 - 31x + 35\)[/tex]
- [tex]\(6x^2 - 31x - 12\)[/tex]
- [tex]\(5x^2 - 21x + 12\)[/tex]
Therefore, the expression [tex]\((3x-5)(2x-7)\)[/tex] is equal to [tex]\(6x^2 - 31x + 35\)[/tex].
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