## anonymous 4 years ago 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

1. Zarkon

18

2. anonymous

@zarkon: could you please explain with steps

3. anonymous

It is just too easy for Zarkon :)

4. Zarkon

you did clasify this as easy :)

5. Mr.Math

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 $$P(-4)=\frac{1}{8}(-4)^2-2(-4)+8=2+8+8=18.$$

6. Zarkon

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

7. Zarkon

used the vertex formula

8. Mr.Math

Oh that's better I think.

9. ash2326

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??

10. Zarkon

you way is good

11. anonymous

Seems I am only one who used Maxima-minima.

12. anonymous

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

13. Mr.Math

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 $$P′(x)=2ax+b=0 \text{ at } x=8.$$

14. Mr.Math

i.e. $$16a+b=0$$.

15. anonymous

Yes Mr.Math you got that right :)

16. Mr.Math

:-)

17. phi

@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

18. ash2326

Thanks Phi, got it :)