To factor the quadratic expression [tex]\( 2x^2 + 11x + 15 \)[/tex], we need to express it as a product of two binomials. Let's go through the steps to achieve this.
1. Identify the coefficients:
The quadratic expression can be written in the standard form [tex]\( ax^2 + bx + c \)[/tex], where
[tex]\[
a = 2, \quad b = 11, \quad c = 15.
\][/tex]
2. Find the product of [tex]\(a\)[/tex] and [tex]\(c\)[/tex]:
[tex]\[
a \cdot c = 2 \cdot 15 = 30.
\][/tex]
3. Determine two numbers that multiply to [tex]\(ac\)[/tex] (30) and add to [tex]\(b\)[/tex] (11):
We need to find two numbers that multiply to 30 and add up to 11. The numbers 5 and 6 fit this requirement:
[tex]\[
5 \cdot 6 = 30 \quad \text{and} \quad 5 + 6 = 11.
\][/tex]
4. Rewrite the middle term using the two numbers:
We can break the middle term [tex]\( 11x \)[/tex] into two terms using 5 and 6:
[tex]\[
2x^2 + 5x + 6x + 15.
\][/tex]
5. Factor by grouping:
Group the terms in pairs and factor out the common factor from each pair:
[tex]\[
(2x^2 + 5x) + (6x + 15).
\][/tex]
From the first group [tex]\( 2x^2 + 5x \)[/tex], factor out [tex]\( x \)[/tex]:
[tex]\[
x(2x + 5).
\][/tex]
From the second group [tex]\( 6x + 15 \)[/tex], factor out 3:
[tex]\[
3(2x + 5).
\][/tex]
Now, we have:
[tex]\[
x(2x + 5) + 3(2x + 5).
\][/tex]
6. Factor out the common binomial factor:
Both terms contain the common factor [tex]\( (2x + 5) \)[/tex], so we can factor this out:
[tex]\[
(2x + 5)(x + 3).
\][/tex]
Thus, the factored form of the quadratic expression [tex]\( 2x^2 + 11x + 15 \)[/tex] is:
[tex]\[
(2x + 5)(x + 3).
\][/tex]
This shows the quadratic expression completely and correctly factored.
