The teacher has written an equation of the form \(p(x)=0\) on the board, where \(p(x)\)is a quadratic, but Heather can't read the linear term. She can see that the quadratic term is \(4x^2\) and that the constant is \(-24\). She asks her neighbor, Noel, what the linear term is.

Noel decides to tease her and just says, ``One of the roots is 4.''

Heather then says, ``Oh, thanks!''

She then correctly writes down the linear term. What was the linear term? Write the entire term, not just the coefficient.

**The answer is NOT 0**

Keihaku Nov 30, 2022

#1**-1 **

Let p(x) = (Ax + B)(Cx + D). Then expanding, we get p(x) = ACx^2 + (AD + BC)x + BD, so AC = 4, BC = -24.

Also, x = 4 is a root, so B = 4A.

Solving this system, we get A = 1, B = 4, C = 4, D = -6, so the constant term is (AD + BC) x = (-6 + 16)x = 10x.

Guest Nov 30, 2022