SOLUTION: Let a and b be the roots of x^2 + 7x - 4 = 0. Find (a + 3)/(b + 3) + (b + 3)/(a + 3).

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(13200) (Show Source): You can put this solution on YOUR website!

Vieta's Theorem tells us that, if a and b are the roots of x^2+7x-4=0, then...

sum of roots = a+b = -7
product of roots = ab = -4

Manipulate the given expression to write it entirely in terms of (a+b), (ab), and constants.

ANSWER: -33/16

Note this solution makes use of an identity that is useful in solving many algebraic problems:

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