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The roots of the equation (x – p) (x – q) = r2, where p, q and r are real, are
- a)always complex
- b)always real
- c)always purely imaginary
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
The roots of the equation (x – p) (x – q) = r2, where p, q...
Solution:
Given equation is (x - p)(x - q) = r^2
Let's expand the equation, we get
x^2 - (p+q)x + pq = r^2
The roots of the equation are given by
x = (p+q)/2 ± √[(p+q)^2/4 - pq + r^2]
Discriminant = [(p+q)^2/4 - pq + r^2]
We need to find for what values of p, q and r, the discriminant is non-negative, so that the roots of the equation are real.
Now, let's analyze the discriminant,
[(p+q)^2/4 - pq + r^2]
= (p^2+q^2)/4 + r^2 - pq
= [(p-q)^2]/4 + r^2
Since, (p-q)^2 ≥ 0 and r^2 ≥ 0, so the discriminant is always non-negative.
Hence, the roots of the given equation are always real.
Therefore, the correct answer is option 'B'.
Given equation is (x - p)(x - q) = r^2
Let's expand the equation, we get
x^2 - (p+q)x + pq = r^2
The roots of the equation are given by
x = (p+q)/2 ± √[(p+q)^2/4 - pq + r^2]
Discriminant = [(p+q)^2/4 - pq + r^2]
We need to find for what values of p, q and r, the discriminant is non-negative, so that the roots of the equation are real.
Now, let's analyze the discriminant,
[(p+q)^2/4 - pq + r^2]
= (p^2+q^2)/4 + r^2 - pq
= [(p-q)^2]/4 + r^2
Since, (p-q)^2 ≥ 0 and r^2 ≥ 0, so the discriminant is always non-negative.
Hence, the roots of the given equation are always real.
Therefore, the correct answer is option 'B'.
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Community Answer
The roots of the equation (x – p) (x – q) = r2, where p, q...
Solution:
Given equation is (x-p)(x-q)=r^2
Let's expand this equation to get a quadratic equation in x:
x^2 - (p+q)x + pq = r^2
Now we have to determine the nature of roots of this quadratic equation.
The discriminant of the quadratic equation is:
D = (p+q)^2 - 4pq - 4r^2
= (p-q)^2 - 4r^2
If D > 0, the roots are real and distinct.
If D = 0, the roots are real and equal.
If D < 0,="" the="" roots="" are="" complex="" conjugates.="" />
Let's consider each case:
Case 1: D > 0
(p-q)^2 - 4r^2 > 0
This implies that (p-q)^2 > 4r^2
Taking square root on both sides, we get:
|p-q| > 2r
This means that the distance between p and q is greater than 2r.
In this case, the roots are real and distinct.
Case 2: D = 0
(p-q)^2 - 4r^2 = 0
This implies that (p-q)^2 = 4r^2
Taking square root on both sides, we get:
|p-q| = 2r
This means that the distance between p and q is equal to 2r.
In this case, the roots are real and equal.
Case 3: D < 0="" />
(p-q)^2 - 4r^2 < 0="" />
This implies that (p-q)^2 < 4r^2="" />
Taking square root on both sides, we get:
|p-q| < 2r="" />
This means that the distance between p and q is less than 2r.
In this case, the roots are complex conjugates.
Therefore, the answer is option 'B', the roots are always real.
Given equation is (x-p)(x-q)=r^2
Let's expand this equation to get a quadratic equation in x:
x^2 - (p+q)x + pq = r^2
Now we have to determine the nature of roots of this quadratic equation.
The discriminant of the quadratic equation is:
D = (p+q)^2 - 4pq - 4r^2
= (p-q)^2 - 4r^2
If D > 0, the roots are real and distinct.
If D = 0, the roots are real and equal.
If D < 0,="" the="" roots="" are="" complex="" conjugates.="" />
Let's consider each case:
Case 1: D > 0
(p-q)^2 - 4r^2 > 0
This implies that (p-q)^2 > 4r^2
Taking square root on both sides, we get:
|p-q| > 2r
This means that the distance between p and q is greater than 2r.
In this case, the roots are real and distinct.
Case 2: D = 0
(p-q)^2 - 4r^2 = 0
This implies that (p-q)^2 = 4r^2
Taking square root on both sides, we get:
|p-q| = 2r
This means that the distance between p and q is equal to 2r.
In this case, the roots are real and equal.
Case 3: D < 0="" />
(p-q)^2 - 4r^2 < 0="" />
This implies that (p-q)^2 < 4r^2="" />
Taking square root on both sides, we get:
|p-q| < 2r="" />
This means that the distance between p and q is less than 2r.
In this case, the roots are complex conjugates.
Therefore, the answer is option 'B', the roots are always real.
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