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p and q are the roots of a quadratic equation. How many such quadratic equations exist for which p and q as well as p2 and q2 are the roots? (Both p and q lie on the positive side of the number-line.)
- a)0
- b)1
- c)2
- d)No such quadratic equation is possible.
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
p and q are the roots of a quadratic equation. How many such quadratic...
A quadratic equation can have only two roots. Hence, we have two cases:
Case 1: p = p2 and q = q2
As the roots are positive numbers, hence the only solution we have is p = q = 1
Case 2: p = q2 and q = p2
We have p = p4
p3 - 1 = 0
As, p is real, p = 1 therefore, q = 1
Hence, from both the cases we get the solution p = q= 1
Hence, only one solution is possible.
Hence, option 2.
Case 1: p = p2 and q = q2
As the roots are positive numbers, hence the only solution we have is p = q = 1
Case 2: p = q2 and q = p2
We have p = p4
p3 - 1 = 0
As, p is real, p = 1 therefore, q = 1
Hence, from both the cases we get the solution p = q= 1
Hence, only one solution is possible.
Hence, option 2.
Most Upvoted Answer
p and q are the roots of a quadratic equation. How many such quadratic...
Solution:
To find the quadratic equations for which p and q as well as p^2 and q^2 are the roots, we need to consider the general form of a quadratic equation:
ax^2 + bx + c = 0
1. Quadratic equation with roots p and q:
If p and q are the roots of the quadratic equation, then we can write the equation as:
a(x - p)(x - q) = 0
Expanding this equation, we get:
ax^2 - a(p+q)x + apq = 0
Comparing the coefficients of this equation with the general form of a quadratic equation, we have:
a = a
b = -a(p+q)
c = apq
2. Quadratic equation with roots p^2 and q^2:
If p^2 and q^2 are the roots of the quadratic equation, then we can write the equation as:
a(x - p^2)(x - q^2) = 0
Expanding this equation, we get:
ax^2 - a(p^2+q^2)x + ap^2q^2 = 0
Comparing the coefficients of this equation with the general form of a quadratic equation, we have:
a = a
b = -a(p^2+q^2)
c = ap^2q^2
3. Equating the coefficients:
To find the quadratic equations for which p and q as well as p^2 and q^2 are the roots, we need to equate the coefficients of both equations.
Comparing the coefficients, we have:
-a(p+q) = -a(p^2+q^2)
apq = ap^2q^2
Simplifying these equations, we get:
p+q = p^2+q^2
pq = p^2q^2
4. Solving the equations:
Let's solve the equations to find the possible values of p and q.
From the first equation, we have:
p+q = p^2+q^2
Rearranging the terms, we get:
p^2 - p + q^2 - q = 0
Simplifying further, we have:
p(p-1) + q(q-1) = 0
Since p and q are positive, this equation implies that p = 1 and q = 0 or vice versa.
From the second equation, we have:
pq = p^2q^2
Since p and q are positive, this equation simplifies to:
1 = p*q
Since p and q are positive integers, the only possible values for p and q are 1 and 1.
Therefore, there is only one quadratic equation for which p and q as well as p^2 and q^2 are the roots. The correct answer is option B.
To find the quadratic equations for which p and q as well as p^2 and q^2 are the roots, we need to consider the general form of a quadratic equation:
ax^2 + bx + c = 0
1. Quadratic equation with roots p and q:
If p and q are the roots of the quadratic equation, then we can write the equation as:
a(x - p)(x - q) = 0
Expanding this equation, we get:
ax^2 - a(p+q)x + apq = 0
Comparing the coefficients of this equation with the general form of a quadratic equation, we have:
a = a
b = -a(p+q)
c = apq
2. Quadratic equation with roots p^2 and q^2:
If p^2 and q^2 are the roots of the quadratic equation, then we can write the equation as:
a(x - p^2)(x - q^2) = 0
Expanding this equation, we get:
ax^2 - a(p^2+q^2)x + ap^2q^2 = 0
Comparing the coefficients of this equation with the general form of a quadratic equation, we have:
a = a
b = -a(p^2+q^2)
c = ap^2q^2
3. Equating the coefficients:
To find the quadratic equations for which p and q as well as p^2 and q^2 are the roots, we need to equate the coefficients of both equations.
Comparing the coefficients, we have:
-a(p+q) = -a(p^2+q^2)
apq = ap^2q^2
Simplifying these equations, we get:
p+q = p^2+q^2
pq = p^2q^2
4. Solving the equations:
Let's solve the equations to find the possible values of p and q.
From the first equation, we have:
p+q = p^2+q^2
Rearranging the terms, we get:
p^2 - p + q^2 - q = 0
Simplifying further, we have:
p(p-1) + q(q-1) = 0
Since p and q are positive, this equation implies that p = 1 and q = 0 or vice versa.
From the second equation, we have:
pq = p^2q^2
Since p and q are positive, this equation simplifies to:
1 = p*q
Since p and q are positive integers, the only possible values for p and q are 1 and 1.
Therefore, there is only one quadratic equation for which p and q as well as p^2 and q^2 are the roots. The correct answer is option B.
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p and q are the roots of a quadratic equation. How many such quadratic equations exist for which p and q as well as p2 and q2 are the roots? (Both p and q lie on the positive side of the number-line.)a)0b)1c)2d)No such quadratic equation is possible.Correct answer is option 'B'. Can you explain this answer?
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