Negative sign in algebraic equations/expressions

GRE Quant Problem Solving
1

I am a bit unclear about negative values and when a negative number replaces a variable in a formula or an equation and goes in a bracket and becomes positive and when it doesn’t. For instance, if we’re given a value for b^2 and that value is -8, I typically put that in a bracket and (-8)^2 becomes 64. But how about when the b value is -b for instance; in the Algebra Progress Quiz 4; we are given this problem x^2 - bx + 16 = 0 and we are told that the expression has only one solution and asked to find the B value.

While I understand how b^2 could be either positive or negative; why do we discount the “-” sign from the original equation in determining the value of b? Does that have no impact on it or am I missing something? Or is the reasoning that when b is -8; if we substitute it in the equation it becomes +8 and when its +8, it essentially becomes -8 when substituted? Sorry this is a bit lengthy and all over the place. I’m quite confused.

2

(-b)^2 -4(1)(16) = 0 but (-b)^2 = (-1)^2 (b^2) = b^2

1 Like
3

Seemed like Latin at first but I think I follow. So what you’re saying in essence is I should treat b^2 as having a co-efficient of negative 1.

4

Do you agree that the discriminant is (-b)^2 - 4(1)(16)?

5

Yes, I do

6

Maybe you can now follow that (-b)^2 = (-1 \cdot b)^2 = (-1)^2 (b)^2= b^2.

To visualize this, consider the graph of y = x^2, which is symmetrical about the y-axis (x = 0). This symmetry shows that a positive number and its negative counterpart are equal when squared.

7

Thank you so much, this is now clear.