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The teacher has written a quadratic on the board, 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? Include entire term, not simply the coefficient

 May 11, 2019
 #1
avatar+17777 
+3

The quadratic has this form:  4x2 + ax - 24  =  0  and the problem is to find the value of a.

Since one of the roots is  4, replace  x  with  4  in the quadratic:

     4x2 + ax - 24  =  0     --->     4(4)2 + a(4) - 24  =  0     --->     4(16) + 4a - 24  =  0     --->     64 + 4a - 24  =  0

                    --->     4a + 40  =  0     --->     4a  =  -40     --->     a  =  -10

So, the linear term is  -10x.

 May 11, 2019
 #2
avatar+105415 
+1

Let the roots be r1  and r2...and let r1  =  4

The product of the roots is

4*r2  =  -24/4  = -6

So  r2  = -6/4   =  -3/2

So....the polynomial  is

 

y = 4 (x - 4)(x + 3/2)   =  4 ( x^2 - 5/2x - 6)  =  4x^2 - 10x - 24

 

So...the linear term is  -10x

 

 

cool cool cool

 May 11, 2019

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