To factor the quadratic expression [tex]\(2x^2 + 13x + 15\)[/tex] completely, follow these steps:
1. Identify the coefficients: The expression is in the standard quadratic form [tex]\(ax^2 + bx + c\)[/tex], where:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = 13\)[/tex]
- [tex]\(c = 15\)[/tex]
2. Find two numbers that multiply to [tex]\(a \times c\)[/tex] and add to [tex]\(b\)[/tex]:
- We need two numbers that multiply to [tex]\(2 \times 15 = 30\)[/tex] and add to [tex]\(13\)[/tex].
- These numbers are [tex]\(3\)[/tex] and [tex]\(10\)[/tex] because [tex]\(3 \times 10 = 30\)[/tex] and [tex]\(3 + 10 = 13\)[/tex].
3. Rewrite the middle term using the two numbers found:
- We can rewrite [tex]\(13x\)[/tex] as [tex]\(3x + 10x\)[/tex].
- The expression thus becomes [tex]\(2x^2 + 3x + 10x + 15\)[/tex].
4. Factor by grouping:
- Group the terms to factor separately: [tex]\( (2x^2 + 3x) + (10x + 15) \)[/tex].
- Factor out the greatest common factor (GCF) from each group:
- The GCF of [tex]\(2x^2 + 3x\)[/tex] is [tex]\(x\)[/tex], so [tex]\(2x^2 + 3x = x(2x + 3)\)[/tex].
- The GCF of [tex]\(10x + 15\)[/tex] is [tex]\(5\)[/tex], so [tex]\(10x + 15 = 5(2x + 3)\)[/tex].
5. Factor out the common binomial:
- Now the expression is [tex]\(x(2x + 3) + 5(2x + 3)\)[/tex].
- Notice that both terms contain the common factor [tex]\((2x + 3)\)[/tex].
- Factor [tex]\((2x + 3)\)[/tex] out: [tex]\((2x + 3)(x + 5)\)[/tex].
So, the completely factored form of the quadratic expression [tex]\(2x^2 + 13x + 15\)[/tex] is:
[tex]\[
(2x + 3)(x + 5)
\][/tex]
