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If a(5+q^(2))+2bq+c=0 and a(5+r^(2))+2br...

If `a(5+q^(2))+2bq+c=0` and `a(5+r^(2))+2br+c=0` where `a ne 0` then q + r =

A

`(-2b)/(a)`

B

`(2b)/(a)`

C

`(a)/(2b)`

D

`(-a)/(2b)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we start with the two equations given: 1. \( a(5 + q^2) + 2bq + c = 0 \) 2. \( a(5 + r^2) + 2br + c = 0 \) Since both equations are equal to zero, we can analyze them as a quadratic equation in terms of a variable \( x \). ### Step 1: Rewrite the equations We can rewrite both equations in a standard quadratic form. Let's denote \( x \) as either \( q \) or \( r \). Thus, we can express the equations as: \[ a(x^2 + 2bx + (5 + c/a)) = 0 \] ### Step 2: Identify the quadratic equation From the equations, we can see that they represent a quadratic equation in \( x \): \[ ax^2 + 2bx + (5a + c) = 0 \] ### Step 3: Use the properties of roots In a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of the roots \( q + r \) can be found using the formula: \[ q + r = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2} = -\frac{2b}{a} \] ### Step 4: Conclusion Thus, we conclude that: \[ q + r = -\frac{2b}{a} \] ### Final Answer The final answer is: \[ q + r = -\frac{2b}{a} \] ---
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