Problem 3 - Beginning and Intermediate Algebra

Question

If \(x^{2}+b x+c\) factors to \((x+m)(x+n)\) and if \(c\) is positive and \(b\) is negative, what do you know about the signs of \(m\) and \(n ?\)

Step-by-Step Solution

Answer

Since the sum \(m+n\) is negative and the product \(mn\) is positive, both \(m\) and \(n\) must have the same sign, which is negative. Therefore, both \(m\) and \(n\) are negative.

Step 1: Write the quadratic equation in the given form

We are given the quadratic equation in the form: \(x^2+bx+c = (x+m)(x+n)\)

Step 2: Equate the coefficients

Now, equate the coefficients of the quadratic: \((x+m)(x+n) = x^2 + (m+n)x +mn\) This gives us: \(m + n = b\) and \(mn = c\)

Step 3: Analyze the sum and product of \(m\) and \(n\)

We know that \(m+n=b\) and this is negative since \(b\) is negative. Also, since \(mn=c\), and \(c\) is positive, then the product \(mn\) is positive as well.

Step 4: Determine the signs of \(m\) and \(n\)

Since their sum is negative and their product is positive, it means that both \(m\) and \(n\) must have the same sign, and it must be negative. Therefore, both \(m\) and \(n\) are negative.