To factor the quadratic expression [tex]\( X^2 - 3X - 15 \)[/tex], follow these steps:
1. Identify the quadratic expression:
The given expression is [tex]\( X^2 - 3X - 15 \)[/tex].
2. Check for common factors:
In this case, there are no common factors between the terms [tex]\( X^2 \)[/tex], [tex]\( -3X \)[/tex], and [tex]\( -15 \)[/tex] other than 1.
3. Apply the quadratic formula method:
While the quadratic formula is an effective method in many cases, factorizing quadratics often requires finding two numbers that multiply to the constant term and add up to the linear coefficient. Here, the coefficients are: [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = -15\)[/tex]. However, the given answer indicates that further factorization might not change the form of the quadratic expression significantly.
4. Verification of Factorization:
Given the answer to this problem is [tex]\( X^2 - 3X - 15 \)[/tex], this suggests that the expression does not factor into simpler binomials without using complex numbers or additional methods, and already is in its simplest quadratic form that concerns factoring over the reals.
Therefore, the factorized form given, which maintains [tex]\( X^2 - 3X - 15 \)[/tex], implies that this is the simplest form given the methods allowed, and no further factoring is achievable within the context expected. Hence, the expression remains:
[tex]\[ X^2 - 3X - 15 \][/tex]
