How would I factor this:

x^2 - 3x - 1

usually we restrict our factors to integer values, or at worst rational numbers.

there are no rational numbers or integers
which when multiplied give me -1 and when added give me -3

so it does not factor

Thanks :)

To factor the quadratic expression \(x^2 - 3x - 1\), we need to find two binomials that, when multiplied, give us the original expression.

Here's how you can do it step by step:

Step 1: Start with the form (x + ?)(x + ?), where ? represents the unknown factors we need to find.

Step 2: The first term of each binomial will be \(x\) since we have \(x^2\) in the original expression.

Step 3: For the second term in each binomial, we need to find two numbers whose product is \(1\) (the coefficient of the constant term) and whose sum is \(-3\) (the coefficient of the linear term).

Step 4: Find the factors of \(1\) that add up to \(-3\). The factors of \(1\) are \(\pm1\) and they add up to \(0\). Unfortunately, no combination of these factors will give us \(-3\).

Step 5: Since we cannot find two numbers that satisfy the requirements in step 4, we conclude that the given quadratic expression \(x^2 - 3x - 1\) is not factorable using integers.

Therefore, the expression \(x^2 - 3x - 1\) cannot be factored further using whole numbers.