Question

Just another problem:

A quadratic polynomial \( P(x) \) is such that \( P(x) \) never takes any negative values and \( P(0)=8 \) and the \( P(8)=0\). Find \( P(-4) \)

Genre: algebra pre-calculus
Rating: Easy

Answer 1

18

Answer 2

@zarkon: could you please explain with steps

Answer 3

It is just too easy for Zarkon :)

Answer 4

you did clasify this as easy :)

Answer 5

Let \(P(x)=ax^2+bx+c\), then \(P(0)=8 \implies c=8\) and
\(P(8)=0 \implies 8^2a+8b+8=0 \implies b=-8a-1\).
Now, \(P(x)=ax^2-(8a+1)x+8\ge 0 \implies (x-8)(ax-1)\ge 0 \implies a=\frac{1}{8}.\)
Thus \(P(-4)=\frac{1}{8}(-4)^2-2(-4)+8=2+8+8=18.\)

Answer 6

i used the fact that (8,0) must be the vertex to get the 2nd equation

Answer 7

used the vertex formula

Answer 8

Oh that's better I think.

Answer 9

Let the quadratic polynomial be
\[P(x)=ax^2+bx+c\]
as it's given it doesn't take negative values
D<0
\[b^2-4ac<0\]
let's substitute x=0 P(0)=8
c=8
P(8)=64a+8b+8=0
or
\[8a+b+1=0\]
we have b^2-4ac<0
or
b^2-32a<0
or
b^2<32 a
we have
8a=-1-b
b^2<-4-4b
b^2+4b+4<0
(b+2)^<0

so b is a complex no. of the form =-2-xi
now let x be 1
so
\[P(x)=\frac{-(1+i)}{8} x^2+(-2+i)x+8\]
Could anyone point out my mistake??

Answer 10

you way is good

Answer 11

Seems I am only one who used Maxima-minima.

Answer 12

maxima-minima : u mean by taking the second derivatives of the eqs and substituting as rt-s^2 ??

Answer 13

We can also use the derivative (I don't know if that what you mean @Fool). As Zarkon said, the polynomial must have its minimum at x=8. So \(P′(x)=2ax+b=0 \text{ at } x=8.\)

Answer 14

i.e. \(16a+b=0\).

Answer 15

Yes Mr.Math you got that right :)

Answer 16

:-)

Answer 17

@ash
if the discriminant is less than zero the square root is imaginary and there are ZERO x-intercepts
however,the problem states P(8)= 0. so we must have a repeated root at x=8, and the discriminant = 0

Zarkon's approach seems to fastest:
given 8 is a root, the equation is
y= a(x-8)^2 + b
0= a(8-8)^2 + b --> b=0
8= a(-8)^2 --> a= 1/8
y= (1/8)*(-4-12)^2 = 144/8 = 18

Answer 18

Thanks Phi, got it :)