SOLUTION: Please solve this problem! I don't understand how to solve it. Given that x^2-2x-5=0 has 2 different roots, and x^2+ax+b=0 has 2 roots, each 2 less than the corresponding root of

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Please solve this problem! I don't understand how to solve it. Given that x^2-2x-5=0 has 2 different roots, and x^2+ax+b=0 has 2 roots, each 2 less than the corresponding root of      Log On


   

Question 988815: Please solve this problem! I don't understand how to solve it.
Given that x^2-2x-5=0 has 2 different roots, and x^2+ax+b=0 has 2 roots, each 2 less than the corresponding root of the equation, find the values of a and b.
(If the roots of x^2-2x-5=0 are alpha and β, the roots of the second equation will be alpha-2 and β-2)
Thank you!!
Answer by josgarithmetic(39614) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2-2x%2B5=0
Formula for General Solution to Quadratic Equation gives
roots %282%2B-+sqrt%28%28-2%29%5E2-4%2A5%29%29%2F2
%282%2B-+sqrt%284-20%29%29%2F2
%282%2B-+sqrt%28-16%29%29%2F2
%282%2B-+4i%29%2F2
highlight%281%2B-+2i%29
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These are roots, 1-2i and 1%2B2i.

The roots for x%5E2%2Bax%2Bb=0 are specified to be
1-2i-2=highlight_green%28-1-2i%29 and 1%2B2i-2=highlight_green%28-1%2B2i%29. You then
want to find solutions for a and b. Again use formula for general solution for x or the roots.
Roots are %28-a%2B-+sqrt%28a%5E2-4b%29%29%2F1
highlight_green%28-a%2B-+sqrt%28a%5E2-4b%29%29

We can compare each expression, but we want an "i", so make this adjustment.
One of the roots is -a-sqrt%28%28-1%29%284b-a%5E2%29%29=-a-i%2Asqrt%284b-a%5E2%29
Other root is -a%2Bsqrt%28%28-1%29%284b-a%5E2%29%29-a%2Bi%2Asqrt%284b-a%5E2%29

Take for convenience, a%3E0 so the signs on a and on the constant 1 will match.
We therefore make the comparisons
system%28-1-2i=-a-i%2Asqrt%284b-a%5E2%29%2C-1%2B2i=-a%2Bi%2Asqrt%284b-a%5E2%29%29
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That permits seeing immediately that because of corresponding terms, highlight%28a=1%29.

Since again by corresponding parts, 2=sqrt%284b-a%5E2%29, we now know a, so just solve this for b.
2=sqrt%284b-1%2A1%29
2=sqrt%284b-1%29
4=4b-1
4%2B1=4b
5=4b
highlight%28b=5%2F4%29


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You might need to reread this posted solution three to six times thinking through carefully to best understand it.