SOLUTION: Find the values of​ a, b, and c for which the quadratic equation ax^2+bx+c=0 has the solutions 5-√26 and 5+ √26. ​(Hint: Use the​ zero-product property in​ reverse.)

Algebra ->  Equations -> SOLUTION: Find the values of​ a, b, and c for which the quadratic equation ax^2+bx+c=0 has the solutions 5-√26 and 5+ √26. ​(Hint: Use the​ zero-product property in​ reverse.)       Log On


   

Question 1192005: Find the values of​ a, b, and c for which the quadratic equation ax^2+bx+c=0 has the solutions 5-√26 and 5+ √26.
​(Hint: Use the​ zero-product property in​ reverse.)
a=
b=
c=
(Type an integer or a simplified faction.)
Thank you so much for your help!(:
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

I'll use a different approach instead of the hint given.
If you prefer the hint route, then let me know and I'll fix my solution.

x+=+5+-+sqrt%2826%29 and x+=+5+%2B+sqrt%2826%29 are the two given roots or x intercepts.

For each equation, subtract 5 from both sides
x-5+=+-sqrt%2826%29 and x-5+=+sqrt%2826%29

If we square both sides, then each right hand side yields 26.
The squaring operation and square root cancel out.
So we have this equation
%28x-5%29%5E2+=+26

Let's get it into standard form ax^2+bx+c = 0
%28x-5%29%5E2+=+26

%28x-5%29%28x-5%29+=+26

x%28x-5%29-5%28x-5%29+=+26

x%5E2-5x-5x%2B25+=+26

x%5E2-5x-5x%2B25-26=0

x%5E2-10x-1=0
Comparing this to ax^2+bx+c = 0, we have
a = 1
b = -10
c = -1
To verify the answers, let's use the quadratic formula to plug in those a,b,c values
We should get the roots your teacher gave you
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-10%29%2B-sqrt%28%28-10%29%5E2-4%281%29%28-1%29%29%29%2F%282%281%29%29

x+=+%2810%2B-sqrt%28100%2B4%29%29%2F%282%29

x+=+%2810%2B-sqrt%28104%29%29%2F%282%29

x+=+%2810%2B-sqrt%284%2A26%29%29%2F%282%29

x+=+%2810%2B-sqrt%284%29%2Asqrt%2826%29%29%2F%282%29

x+=+%2810%2B-++++2%2Asqrt%2826%29%29%2F%282%29

x+=+%282%285%2B-sqrt%2826%29%29%29%2F%282%29

x+=+5%2B-sqrt%2826%29

x+=+5%2Bsqrt%2826%29 or x+=+5-sqrt%2826%29
We've confirmed the answers.

Answers:
a = 1
b = -10
c = -1

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


We can assume at the start that a=1; if that leads to values of b and/or c that are not integers, we can scale up our answer.

If a is 1, then b is the opposite of the sum of the roots, and c is the product of the roots.

The sum of the roots is 10.

The product of the roots is 25-26=-1.

ANSWERS:
a=1
b=-10
c=-1