Answer :
We want to factor the quadratic expressions
[tex]$$6x^2 + 37x + 6$$[/tex]
and
[tex]$$x^2 + 37x + 36.$$[/tex]
We'll work through each one step by step.
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Step 1. Factorizing [tex]$$6x^2 + 37x + 6$$[/tex]
1.1 Identify the coefficients:
[tex]$$a = 6, \quad b = 37, \quad c = 6.$$[/tex]
1.2 Compute the product of the leading coefficient and the constant term:
[tex]$$a \cdot c = 6 \times 6 = 36.$$[/tex]
1.3 Find two numbers that multiply to 36 and add to 37.
The numbers are 36 and 1 because
[tex]$$36 \times 1 = 36 \quad \text{and} \quad 36 + 1 = 37.$$[/tex]
1.4 Rewrite the middle term using these two numbers:
[tex]$$6x^2 + 36x + x + 6.$$[/tex]
1.5 Factor by grouping:
Group the terms as follows:
[tex]$$ (6x^2 + 36x) + (x + 6).$$[/tex]
1.6 Factor out the greatest common factor in each group:
From the first group, factor out [tex]$6x$[/tex]:
[tex]$$6x^2 + 36x = 6x(x + 6).$$[/tex]
From the second group, factor out [tex]$1$[/tex]:
[tex]$$x + 6 = 1(x + 6).$$[/tex]
1.7 Now, factor out the common binomial [tex]$(x + 6)$[/tex]:
[tex]$$6x(x + 6) + 1(x + 6) = (x + 6)(6x + 1).$$[/tex]
Thus, the factorization is:
[tex]$$6x^2 + 37x + 6 = (6x + 1)(x + 6).$$[/tex]
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Step 2. Factorizing [tex]$$x^2 + 37x + 36$$[/tex]
2.1 Identify the coefficients for the monic quadratic (where [tex]$a = 1$[/tex]):
[tex]$$a = 1, \quad b = 37, \quad c = 36.$$[/tex]
2.2 Since [tex]$a = 1$[/tex], we only need to find two numbers that multiply to [tex]$c$[/tex] and add to [tex]$b$[/tex].
Here, we look for two numbers that multiply to
[tex]$$1 \times 36 = 36$$[/tex]
and add to 37.
The numbers are 1 and 36 because
[tex]$$1 \times 36 = 36 \quad \text{and} \quad 1 + 36 = 37.$$[/tex]
2.3 Rewrite the middle term using these two numbers:
[tex]$$x^2 + 1x + 36x + 36.$$[/tex]
2.4 Factor by grouping:
Group the terms as follows:
[tex]$$ (x^2 + x) + (36x + 36).$$[/tex]
2.5 Factor out the greatest common factor in each group:
From the first group, factor out [tex]$x$[/tex]:
[tex]$$x^2 + x = x(x + 1).$$[/tex]
From the second group, factor out [tex]$36$[/tex]:
[tex]$$36x + 36 = 36(x + 1).$$[/tex]
2.6 Now, factor out the common binomial [tex]$(x + 1)$[/tex]:
[tex]$$x(x + 1) + 36(x + 1) = (x + 1)(x + 36).$$[/tex]
Thus, the factorization is:
[tex]$$x^2 + 37x + 36 = (x + 1)(x + 36).$$[/tex]
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Final Answers:
1. For [tex]$$6x^2 + 37x + 6:$$[/tex]
[tex]$$ (6x + 1)(x + 6) $$[/tex]
2. For [tex]$$x^2 + 37x + 36:$$[/tex]
[tex]$$ (x + 1)(x + 36) $$[/tex]
These steps show how each quadratic was factored systematically using the technique of splitting the middle term and grouping.
[tex]$$6x^2 + 37x + 6$$[/tex]
and
[tex]$$x^2 + 37x + 36.$$[/tex]
We'll work through each one step by step.
──────────────────────────────
Step 1. Factorizing [tex]$$6x^2 + 37x + 6$$[/tex]
1.1 Identify the coefficients:
[tex]$$a = 6, \quad b = 37, \quad c = 6.$$[/tex]
1.2 Compute the product of the leading coefficient and the constant term:
[tex]$$a \cdot c = 6 \times 6 = 36.$$[/tex]
1.3 Find two numbers that multiply to 36 and add to 37.
The numbers are 36 and 1 because
[tex]$$36 \times 1 = 36 \quad \text{and} \quad 36 + 1 = 37.$$[/tex]
1.4 Rewrite the middle term using these two numbers:
[tex]$$6x^2 + 36x + x + 6.$$[/tex]
1.5 Factor by grouping:
Group the terms as follows:
[tex]$$ (6x^2 + 36x) + (x + 6).$$[/tex]
1.6 Factor out the greatest common factor in each group:
From the first group, factor out [tex]$6x$[/tex]:
[tex]$$6x^2 + 36x = 6x(x + 6).$$[/tex]
From the second group, factor out [tex]$1$[/tex]:
[tex]$$x + 6 = 1(x + 6).$$[/tex]
1.7 Now, factor out the common binomial [tex]$(x + 6)$[/tex]:
[tex]$$6x(x + 6) + 1(x + 6) = (x + 6)(6x + 1).$$[/tex]
Thus, the factorization is:
[tex]$$6x^2 + 37x + 6 = (6x + 1)(x + 6).$$[/tex]
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Step 2. Factorizing [tex]$$x^2 + 37x + 36$$[/tex]
2.1 Identify the coefficients for the monic quadratic (where [tex]$a = 1$[/tex]):
[tex]$$a = 1, \quad b = 37, \quad c = 36.$$[/tex]
2.2 Since [tex]$a = 1$[/tex], we only need to find two numbers that multiply to [tex]$c$[/tex] and add to [tex]$b$[/tex].
Here, we look for two numbers that multiply to
[tex]$$1 \times 36 = 36$$[/tex]
and add to 37.
The numbers are 1 and 36 because
[tex]$$1 \times 36 = 36 \quad \text{and} \quad 1 + 36 = 37.$$[/tex]
2.3 Rewrite the middle term using these two numbers:
[tex]$$x^2 + 1x + 36x + 36.$$[/tex]
2.4 Factor by grouping:
Group the terms as follows:
[tex]$$ (x^2 + x) + (36x + 36).$$[/tex]
2.5 Factor out the greatest common factor in each group:
From the first group, factor out [tex]$x$[/tex]:
[tex]$$x^2 + x = x(x + 1).$$[/tex]
From the second group, factor out [tex]$36$[/tex]:
[tex]$$36x + 36 = 36(x + 1).$$[/tex]
2.6 Now, factor out the common binomial [tex]$(x + 1)$[/tex]:
[tex]$$x(x + 1) + 36(x + 1) = (x + 1)(x + 36).$$[/tex]
Thus, the factorization is:
[tex]$$x^2 + 37x + 36 = (x + 1)(x + 36).$$[/tex]
──────────────────────────────
Final Answers:
1. For [tex]$$6x^2 + 37x + 6:$$[/tex]
[tex]$$ (6x + 1)(x + 6) $$[/tex]
2. For [tex]$$x^2 + 37x + 36:$$[/tex]
[tex]$$ (x + 1)(x + 36) $$[/tex]
These steps show how each quadratic was factored systematically using the technique of splitting the middle term and grouping.
