SOLUTION: Please solve: Given that the root of ax^2+bx+2=0 is 2-{{{sqrt(2)}}}, find the values of a and b, and the other root. (assume a and b are integers) Thanks.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Please solve: Given that the root of ax^2+bx+2=0 is 2-{{{sqrt(2)}}}, find the values of a and b, and the other root. (assume a and b are integers) Thanks.       Log On


   

Question 985658: Please solve:
Given that the root of ax^2+bx+2=0 is 2-sqrt%282%29, find the values of a and b, and the other root. (assume a and b are integers)
Thanks.
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!

One particular solution is the quadratic polynomial  x%5E2+-+4x+%2B+2 
and the quadratic equation

x%5E2+-4x+%2B+2 = 0.

It has integer coefficients and the roots  2+%2B+sqrt%282%29  and  2+-+sqrt%282%29.

Any other quadratic polynomial with the multiple coefficients,  like

mx%5E2+-4mx+%2B+2m

with an integer  m  and the corresponding quadratic equation

mx%5E2+-4mx+%2B+2m = 0

is also the solution.


Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!

Please solve:
Given that the root of ax^2+bx+2=0 is 2-sqrt%282%29, find the values of a and b, and the other root. (assume a and b are integers)
Thanks.
Since one root is: 2+-+sqrt%282%29, then other root is: highlight_green%282+%2B+sqrt%282%29%29

Sum of roots: 2+-+sqrt%282%29+%2B+2+%2B+sqrt%282%29 = 4

Sum of roots: -+b%2Fa, so -+b%2Fa+=+4 --------- eq (i)



Product of roots: %282+-+sqrt%282%29%29%282+%2B+sqrt%282%29%29 -----> 4 - 2, or 2

Product of roots: c%2Fa, so c%2Fa+=+2_____2%2Fa+=+2_____2a+=+2______highlight_green%28a+=+1%29 

-+b%2F1+=+4 ---------- Substituting 1 for a in eq (i)

- b = 4 ---------- Cross-multiplying

highlight_green%28b+=+-+4%29