Algebra Hard Practice Questions-#4 page 164 and 170

[sakane808]

I believe there is an error for #4 on page 170. The explanations contradicts the answers.

[tommywallach]

Hey Sakana,

I looked it over and I don't see any problem. It says the answers are A and B, and in the explanation, it says that A and B are correct and C is incorrect. Am I missing something? Perhaps give more of an explanation?

-t

[sakane808]

I guess I'm confused as to why C is incorrect and B is correct. Could you explain the purpose of the different cases when the book provided an equation.

[tommywallach]

Hey Sakane,

The explanation in the book is solid. I would encourage you to try plugging in numbers to work it out for yourself. However, I can also do it here theoretically (but in the future, it's best for your practice to actually try to work it out yourself with real numbers). So here's the theory:

If a > 0, that means it really will be a NEGATIVE value for the coefficient "a," because we already have a negative sign there. Similarly, the constant b will be negative, because we have a plus sign before the b. So the final will be more like this:

x^2 -ax - b = 0 (to the extent that we now know that both the a and the b should have that minus sign in front)

Anytime the constant "b" is negative, it means our solutions have opposite signs, because that "b" represents the product of the two things that we put in parentheses when we factor out a quadratic. This means A is true.

For the sum, we know that a is negative. A represents the sum of the two things that go in parentheses. This would lead us to believe that the sum is less than zero. HOWEVER, the question is asking us about the sum of the SOLUTIONS, which would be the opposite of what's in the parentheses. Let me show you:

Imagine we had x^2 - 6x -16

This would factor to (x - 8) (x + 2)

Now (-8) + 2 is -6, which is less than zero.

But the solutions here would be 8 and -2, the sum of which is 6, which is greater than zero. This means B is true.

Finally, you can see from the example I just gave that the product of my two solutions (-8 and 2) would be -16. This is b, not -b. So C is not true.

Hope that helps!

-t

[hhussain85]

Thanks for your reply, Tom!

It clarified a lot.

Messed up with the b and -b.

[tommywallach]

Glad to help! It's a confusing question.