SOLUTION: Form quadratic functions having the following properties: Zeroes at 1ħsqrt(2), y-intercept at –4. (I tried the answers 4x^2-8x-4 and -4x^2-8x-4, but it didn't work) Th

Algebra ->  Rational-functions -> SOLUTION: Form quadratic functions having the following properties: Zeroes at 1ħsqrt(2), y-intercept at –4. (I tried the answers 4x^2-8x-4 and -4x^2-8x-4, but it didn't work) Th      Log On


   

Question 1071386: Form quadratic functions having the following properties:

Zeroes at 1ħsqrt(2), y-intercept at –4.

(I tried the answers 4x^2-8x-4 and -4x^2-8x-4, but it didn't work)
Thank you
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
Given f%28x%29=4x%5E2-8x-4 as one of multiple choices,
you should find it works because
1) the y-intercept is f%280%29=-4 , and
system%28f%28x%29=4x%5E2-8x-4%2Cf%28x%29=0%29-->4x%5E2-8x-4=0-->x%5E2-2x-1=0-->x%5E2-2x=1-->x%5E2-2x%2B1=1%2B1-->%28x-1%29%5E2=2-->x=1+%2B-+sqrt%282%29

Given no multiple choice answers, you would have to find
function f%28x%29 yourself.

ONE WAY TO DO IT:
With zeros at 1+%2B-+sqrt%282%29 ,
the factored form of the quadratic function would be
f%28x%29=a%28x-%281+%2B+sqrt%282%29%29%29%28x-%281+-+sqrt%282%29%29%29
Simplifying and multiplying we get the standard f%28x%29=ax%5E2%2Bbx%2Bc form.
f%28x%29=a%28x-1+-+sqrt%282%29%29%28x-1+%2B+sqrt%282%29%29
f%28x%29=a%28%28x-1%29%5E2+-+%28sqrt%282%29%29%5E2%29
f%28x%29=a%28%28x%5E2-2x%2B1%29+-+2%29
f%28x%29=a%28x%5E2-2x%2B1+-+2%29
f%28x%29=a%28x%5E2-2x-1%29
f%28x%29=ax%5E2-2ax-a
The y-intercept is
system%28f%280%29=-4%2Cf%280%29=++-a%29 , so -4=-a --> a=4 ,
and f%28x%29=4x%5E2-2%2A4x-4 --> f%28x%29=4x%5E2-8x-4
graph%28300%2C300%2C-1%2C3%2C-10%2C10%2C4x%5E2-8x-4%29

ANOTHER WAY:
f%28x%29=ax%5E2%2Bbx%2Bc with a%3C%3E0 represents any and all quadratic functions.
If -4 is the y-intercept,
system%28f%280%29=-4%2Cf%280%29=a%2Ao%5E2%2Bb%2A0%2Bc%29 --> system%28f%280%29=-4%2Cf%280%29=c%29 -->highlight%28c=-4%29
The quadratic formula says that the zeros are given by

If those zeros are
x+=+1+%2B-+sqrt%282%29+ , then setting the rational and irrational parts as equal,
system%28%28-b%29%2F2a+=1%2C+sqrt%28+b%5E2-4%2Aa%2A%28-4%29%29%2Fabs%282a%29=sqrt%282%29%29-->system%28b=-2a+%2C+%28b%5E2%2B16a%29%2F4a%5E2=2%29-->system%28b=-2a%2C%284a%5E2%2B16a%29%2F4a%5E2=2%29-->system%28b=-2a%2C4a%5E2%2B16a=2%2A4a%5E2%29-->system%28b=-2a%2C4a%5E2%2B16a=8a%5E2%29-->system%28b=-2a%2C16a=8a%5E2-4a%5E2%29-->system%28b=-2a%2C16a=4a%5E2%29-->system%28b=-2a%2C16=4a%29-->system%28b=-2a%2C16%2F4=a%29-->system%28b=-2a%2Ca=4%29-->system%28b=-2%2A4%2Ca=4%29-->highlight%28system%28b=-8%2Ca=4%29%29 .