Class 10 Exam > Class 10 Questions > If the equation2x2 + 2(c - 20) x + (c - 20) =...
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If the equation 2x2 + 2(c - 20) x + (c - 20) = 0 has equal roots then, the value of c is
- a)20, 22
- b)22, 24
- c)22
- d)20
Correct answer is option 'A'. Can you explain this answer?
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If the equation2x2 + 2(c - 20) x + (c - 20) = 0 has equal roots then, ...
Given equation: 2x^2 + 2(c - 20)x + (c - 20) = 0
To find the value of c for which the equation has equal roots, we need to use the discriminant of the quadratic equation.
The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
If the discriminant is equal to zero (D = 0), then the equation has equal roots.
Let's apply this to the given equation:
Coefficient of x^2 = 2a = 2
Coefficient of x = 2(c - 20) = 2c - 40
Constant term = (c - 20)
Using the formula for the discriminant, we have:
D = (2(c - 20))^2 - 4(2)((c - 20))
= 4(c - 20)^2 - 8(c - 20)
= 4(c^2 - 40c + 400) - 8(c - 20)
= 4c^2 - 160c + 1600 - 8c + 160
= 4c^2 - 168c + 1760
For the given equation to have equal roots, the discriminant D should be equal to zero.
Therefore, we have:
4c^2 - 168c + 1760 = 0
Simplifying the equation, we divide by 4:
c^2 - 42c + 440 = 0
Now, we need to factorize the quadratic equation to find its roots.
(c - 20)(c - 22) = 0
Setting each factor equal to zero, we get:
c - 20 = 0 or c - 22 = 0
Solving these equations, we find:
c = 20 or c = 22
Therefore, the value of c for which the equation has equal roots is c = 20, 22.
Hence, the correct answer is option 'A': 20, 22.
To find the value of c for which the equation has equal roots, we need to use the discriminant of the quadratic equation.
The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
If the discriminant is equal to zero (D = 0), then the equation has equal roots.
Let's apply this to the given equation:
Coefficient of x^2 = 2a = 2
Coefficient of x = 2(c - 20) = 2c - 40
Constant term = (c - 20)
Using the formula for the discriminant, we have:
D = (2(c - 20))^2 - 4(2)((c - 20))
= 4(c - 20)^2 - 8(c - 20)
= 4(c^2 - 40c + 400) - 8(c - 20)
= 4c^2 - 160c + 1600 - 8c + 160
= 4c^2 - 168c + 1760
For the given equation to have equal roots, the discriminant D should be equal to zero.
Therefore, we have:
4c^2 - 168c + 1760 = 0
Simplifying the equation, we divide by 4:
c^2 - 42c + 440 = 0
Now, we need to factorize the quadratic equation to find its roots.
(c - 20)(c - 22) = 0
Setting each factor equal to zero, we get:
c - 20 = 0 or c - 22 = 0
Solving these equations, we find:
c = 20 or c = 22
Therefore, the value of c for which the equation has equal roots is c = 20, 22.
Hence, the correct answer is option 'A': 20, 22.
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