Find the value of k for which the Quadratic equation 4x2 – 2 (k+1)x + (k+1) was equal roots.
Answers
The condition says that the discriminant of a quadratic equation must be zero if one has equal roots, i.e., if and only if the equation ax² + bx + c = 0 has equal roots, then b² - 4ac = 0.
Here we have the equation 4x² - 2(k + 1)x + (k + 1) = 0. Since the equation has equal roots,
[-2(k + 1)]² - [4 × 4(k + 1)] = 0
4(k + 1)² - 16(k + 1) = 0
4(k + 1)(k + 1 - 4) = 0
4(k + 1)(k - 3) = 0
Thus,
k = -1 ; k = 3
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Question:
Find the value of k for which the quadratic equation 4x² - 2(k+1)x + (k+1) = 0 has equal roots.
Answer:
k = -1 , 3
Note:
• An equation of degree 2 is know as quadratic equation .
• Roots of an equation is defined as the possible values of the unknown (variable) for which the equation is satisfied.
• The maximum number of roots of an equation will be equal to its degree.
• A quadratic equation has atmost two roots.
• The general form of a quadratic equation is given as , ax² + bx + c = 0 .
• A quadratic equation has atmost two roots .
• The discriminant of the quadratic equation is given as , D = b² - 4ac .
• If D = 0 , then the quadratic equation would have real and equal roots .
• If D > 0 , then the quadratic equation would have real and distinct roots .
• If D < 0 , then the quadratic equation would have imaginary roots .
Solution:
The given quadratic equation is ;
4x² - 2(k+1)x + (k+1) = 0
Clearly , we have ;
a = 4
b = -2(k+1)
c = (k+1)
We know that ,
The quadratic equation will have equal roots if its discriminant is equal to zero .
=> D = 0
=> [-2(k+1)]² - 4•4•(k+1) = 0
=> 4(k+1)² - 4•4(k+1) = 0
=> 4(k+1)•(k+1-4) = 0
=> (k+1)(k-3) = 0
=> k = -1,3