To factor the quadratic expression [tex]\( x^2 + 12x + 35 \)[/tex], we need to express it as a product of two binomials. Here is the detailed, step-by-step solution:
1. Identify the quadratic expression:
[tex]\[
x^2 + 12x + 35
\][/tex]
2. Look for two numbers that multiply to the constant term (35) and add up to the coefficient of the linear term (12).
- Consider the constant term: 35.
- Consider the coefficient of the linear term: 12.
3. Find the pair of factors of 35:
- The factors of 35 are: 1 and 35, 5 and 7.
4. Determine which pair of factors adds up to 12:
- [tex]\(1 + 35 = 36\)[/tex] (does not work)
- [tex]\(5 + 7 = 12\)[/tex] (works)
So, the numbers 5 and 7 are our factors.
5. Rewrite the middle term (12x) of the quadratic expression using these factors:
[tex]\[
x^2 + 12x + 35 = x^2 + 5x + 7x + 35
\][/tex]
6. Group the terms to simplify the expression:
[tex]\[
x^2 + 5x + 7x + 35 = (x^2 + 5x) + (7x + 35)
\][/tex]
7. Factor out the common factors from each group:
[tex]\[
(x^2 + 5x) + (7x + 35) = x(x + 5) + 7(x + 5)
\][/tex]
8. Factor out the common binomial factor [tex]\((x + 5)\)[/tex]:
[tex]\[
x(x + 5) + 7(x + 5) = (x + 5)(x + 7)
\][/tex]
Therefore, the factored form of the quadratic expression [tex]\( x^2 + 12x + 35 \)[/tex] is:
[tex]\[
(x + 5)(x + 7)
\][/tex]
