To factor the quadratic expression [tex]\( x^2 - x - 56 \)[/tex], follow these steps:
1. Identify the coefficients:
- The quadratic expression is [tex]\( ax^2 + bx + c \)[/tex].
- Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -1 \)[/tex], and [tex]\( c = -56 \)[/tex].
2. Determine the factors of the constant term [tex]\( c \)[/tex]:
- We need to find two numbers that multiply to give [tex]\( c = -56 \)[/tex] and add to give [tex]\( b = -1 \)[/tex].
3. List the factor pairs of [tex]\( -56 \)[/tex]:
- [tex]\((-1, 56)\)[/tex]
- [tex]\((1, -56)\)[/tex]
- [tex]\((-2, 28)\)[/tex]
- [tex]\((2, -28)\)[/tex]
- [tex]\((-4, 14)\)[/tex]
- [tex]\((4, -14)\)[/tex]
- [tex]\((-7, 8)\)[/tex]
- [tex]\((7, -8)\)[/tex]
4. Find the correct pair of factors:
- We need to find a pair that adds up to [tex]\( -1 \)[/tex].
- Among them, [tex]\((7, -8)\)[/tex] satisfies this condition because [tex]\( 7 + (-8) = -1 \)[/tex].
5. Write the quadratic expression using these factors:
- The factors of [tex]\( x^2 - x - 56 \)[/tex] are [tex]\( (x + 7) \)[/tex] and [tex]\( (x - 8) \)[/tex].
6. Write out the factored form:
- The factored form of [tex]\( x^2 - x - 56 \)[/tex] is [tex]\( (x - 8)(x + 7) \)[/tex].
However, your options also need to be checked correctly. The correct factored forms given in the options are:
- [tex]\((x - 8)(x + 7)\)[/tex]
- [tex]\((x - 7)(x + 8)\)[/tex]
- [tex]\((x - 28)(x + 2)\)[/tex]
- [tex]\((x - 2)(x + 28)\)[/tex]
Our factored form is [tex]\((x - 8)(x + 7)\)[/tex]. Reviewing the options, we confirm that:
- Our factorization precisely matches the first option [tex]\((x - 8)(x + 7)\)[/tex].
So, the completely factored form of [tex]\( x^2 - x - 56 \)[/tex] is:
[tex]\[ (x - 8)(x + 7) \][/tex]
And the correct answer is:
[tex]\[ (x - 8)(x + 7) \][/tex]
