Question

suppose you can factor x^2+bx+c as (x+p)(x+q). If c>0, what could be possible values of p and q?

Answer 1

[AI-generated answer — Bot]

If c>0, then p and q must both be negative. Therefore, possible values of p and q are -p and -q, where p and q are any real numbers.

Answer 2

also they could both be positive

Answer 3

[AI-generated answer — Step-by-Step Bot]

To factor the quadratic expression x^2 + bx + c as (x + p)(x + q), we need to find values for p and q. Given that c > 0, we can consider the possible cases:

1. If c is a positive number, p and q can be either both positive or both negative.

2. If c is a negative number, p and q must have opposite signs – one positive and one negative.

Therefore, the possible values of p and q are as follows:

1. If c > 0:
- p > 0, q > 0 (both positive)
- p < 0, q < 0 (both negative)

2. If c < 0:
- p > 0, q < 0 (opposite signs)
- p < 0, q > 0 (opposite signs)

In all cases, p and q should be such that when multiplied together, they give the value of c.

Answer 4

[AI-generated answer — Explain Bot]

To determine the values of p and q when factoring the expression x^2 + bx + c as (x + p)(x + q), where c > 0, we need to use the following information from the quadratic equation:

1. The sum of the roots is -b/a.
2. The product of the roots is c/a.

Considering that c > 0, we know that both roots must have the same sign (either both positive or both negative) since their product is positive.

Given this, let's analyze the possible cases for p and q:

Case 1: p > 0 and q > 0
In this case, both roots are positive, and p and q are positive numbers.

Case 2: p < 0 and q < 0
In this case, both roots are negative, and p and q are negative numbers.

It is important to note that the signs of p and q can be interchanged. In other words, if p is positive, q can be negative, and vice versa. However, the product of p and q should remain the same.

So, the possible values for p and q when factoring x^2 + bx + c as (x + p)(x + q), where c > 0, are:
1. p > 0 and q > 0
2. p < 0 and q < 0

Remember to consider the specific values of b and c in the quadratic equation to determine the exact values of p and q.