Question 1209123: Let a and b be the roots of x^2 + 7x - 4 = 0. Find (a + 3)/(b + 3) + (b + 3)/(a + 3).
Found 2 solutions by math_tutor2020, greenestamps:
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Answer: -33/16
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Explanation
I'll use p,q in place of a,b
This is because a,b,c are the standard coefficients of the quadratic template  .
In the case of x^2+7x-4 = 0 we have a = 1, b = 7, c = -4.
Instead of computing  the expression I'll evaluate is 
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I'll take a slight detour for a moment.
From the quadratic version of Vieta's Formulas, we know that:
p+q = -b/a
p*q = c/a
When plugging a = 1, b = 7, and c = -4, we get
p+q = -b/a = -7/1 = -7
p*q = c/a = -4/1 = -4
In short,
p+q = -7
p*q = -4
Let's call these equation (1) and equation (2) to be used later.
Then note the following
 Applying equations (1) and (2)
 Let's call this equation (3)
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Let's return to
We'll combine the fractions.
Recall we need the LCD to do so.
= 
= 
= 
= 
= 
=  Apply equations (1) through (3)
= 
= 
= 
Therefore,
where p,q are the roots of 
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To verify, you can use the quadratic formula to solve
You should get  and  as the two roots.
Then plug each value into and simplify.
I used GeoGebra to verify the answer.
Here's the link to that calculation
https://www.geogebra.org/calculator/fwzwpynj
Let me know if you have any questions.
Answer by greenestamps(13198)  (Show Source):
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