Question

current objective
factor a trinomial of the form x^2+bx+c whe

question
factor: \\(x^2 - x - 42\\).

provide your answer below:

Explanation:

Identify the trinomial coefficients

We need to factor the quadratic trinomial:

$$x^2 - x - 42$$

Here, the coefficients are:

$$a = 1, \quad b = -1, \quad c = -42$$

Find two numbers that multiply to c and add to b

We look for two integers, \(p\) and \(q\), such that:

$$p \cdot q = -42$$

$$p + q = -1$$

Since the product is negative, one number must be positive and the other negative. Since their sum is \(-1\), the negative number must have a larger absolute value by \(1\).
Testing factors of \(42\):

$$6 \cdot (-7) = -42$$

$$6 + (-7) = -1$$

Thus, the two numbers are \(6\) and \(-7\).

Rewrite and factor by grouping

Using the Factoring by Grouping knowledge point

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Answer:

Factor: \(x^2 - x - 42\) = <blank>\((x - 7)(x + 6)\)</blank>