To factorize the quadratic equation [tex]\(x^2 - x - 42\)[/tex], we follow these steps:
1. Identify the quadratic expression: The given quadratic expression is [tex]\(x^2 - x - 42\)[/tex].
2. Find two numbers that multiply to the constant term (-42) and add up to the coefficient of the linear term (-1):
- The constant term in the quadratic expression is [tex]\(-42\)[/tex].
- The coefficient of the linear term is [tex]\(-1\)[/tex].
- We need to find two numbers, [tex]\(a\)[/tex] and [tex]\(b\)[/tex], such that [tex]\(a \cdot b = -42\)[/tex] and [tex]\(a + b = -1\)[/tex].
3. Determine the pair of numbers: The numbers that satisfy these conditions are [tex]\(-7\)[/tex] and [tex]\(6\)[/tex] because:
- [tex]\(-7 \times 6 = -42\)[/tex]
- [tex]\(-7 + 6 = -1\)[/tex]
4. Write the quadratic expression in factorized form: Using these numbers, we can write the quadratic expression as a product of two binomials:
[tex]\[ x^2 - x - 42 = (x - 7)(x + 6) \][/tex]
Therefore, the factorized form of the quadratic expression [tex]\(x^2 - x - 42\)[/tex] is:
[tex]\[ (x - 7)(x + 6) \][/tex]
