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Answer :
To factor the quadratic expression [tex]\(35x^2 - 29x + 6\)[/tex], we'll break it down step-by-step:
1. Check if it's Factorable:
- We need to find two numbers that multiply to the product of the leading coefficient (35) and the constant term (6), which is [tex]\(35 \times 6 = 210\)[/tex].
- These two numbers should also add up to the middle coefficient, which is [tex]\(-29\)[/tex].
2. Find Suitable Pair:
- After checking various combinations, we find that the numbers [tex]\(-14\)[/tex] and [tex]\(-15\)[/tex] multiply to 210 and add to [tex]\(-29\)[/tex].
3. Split the Middle Term:
- Rewrite the expression by splitting the middle term [tex]\(-29x\)[/tex] into two terms using the numbers found: [tex]\(-14x\)[/tex] and [tex]\(-15x\)[/tex].
- The expression becomes:
[tex]\[
35x^2 - 14x - 15x + 6
\][/tex]
4. Factor by Grouping:
- Group the terms in pairs:
[tex]\[
(35x^2 - 14x) + (-15x + 6)
\][/tex]
- Factor out the greatest common factor from each pair:
- From the first pair, factor out [tex]\(7x\)[/tex]:
[tex]\[
7x(5x - 2)
\][/tex]
- From the second pair, factor out [tex]\(-3\)[/tex]:
[tex]\[
-3(5x - 2)
\][/tex]
5. Combine the Groups:
- Now, notice that both terms contain the common factor [tex]\((5x - 2)\)[/tex], so we can factor this out:
[tex]\[
(7x - 3)(5x - 2)
\][/tex]
So, the expression [tex]\(35x^2 - 29x + 6\)[/tex] factors to [tex]\((5x - 2)(7x - 3)\)[/tex].
1. Check if it's Factorable:
- We need to find two numbers that multiply to the product of the leading coefficient (35) and the constant term (6), which is [tex]\(35 \times 6 = 210\)[/tex].
- These two numbers should also add up to the middle coefficient, which is [tex]\(-29\)[/tex].
2. Find Suitable Pair:
- After checking various combinations, we find that the numbers [tex]\(-14\)[/tex] and [tex]\(-15\)[/tex] multiply to 210 and add to [tex]\(-29\)[/tex].
3. Split the Middle Term:
- Rewrite the expression by splitting the middle term [tex]\(-29x\)[/tex] into two terms using the numbers found: [tex]\(-14x\)[/tex] and [tex]\(-15x\)[/tex].
- The expression becomes:
[tex]\[
35x^2 - 14x - 15x + 6
\][/tex]
4. Factor by Grouping:
- Group the terms in pairs:
[tex]\[
(35x^2 - 14x) + (-15x + 6)
\][/tex]
- Factor out the greatest common factor from each pair:
- From the first pair, factor out [tex]\(7x\)[/tex]:
[tex]\[
7x(5x - 2)
\][/tex]
- From the second pair, factor out [tex]\(-3\)[/tex]:
[tex]\[
-3(5x - 2)
\][/tex]
5. Combine the Groups:
- Now, notice that both terms contain the common factor [tex]\((5x - 2)\)[/tex], so we can factor this out:
[tex]\[
(7x - 3)(5x - 2)
\][/tex]
So, the expression [tex]\(35x^2 - 29x + 6\)[/tex] factors to [tex]\((5x - 2)(7x - 3)\)[/tex].
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