Advertisements
Advertisements
Question
Find the values of k for which the roots are real and equal in each of the following equation:
(3k+1)x2 + 2(k + 1)x + k = 0
Advertisements
Solution
The given quadric equation is (3k+1)x2 + 2(k + 1)x + k = 0, and roots are real and equal
Then find the value of k.
Here, a = (3k + 1), b = 2(k + 1) and c = k
As we know that D = b2 - 4ac
Putting the value of a = (3k + 1), b = 2(k + 1) and c = k
= (2(k + 1))2 - 4 x (3k + 1) x (k)
= 4(k2 + 2k + 1) - 4k(3k + 1)
= 4k2 + 8k + 4 - 12k2 - 4k
= -8k2 + 4k + 4
The given equation will have real and equal roots, if D = 0
Thus,
-8k2 + 4k + 4 = 0
-4(2k2 - k - 1) = 0
2k2 - k - 1 = 0
Now factorizing of the above equation
2k2 - k - 1 = 0
2k2 - 2k + k - 1 = 0
2k(k - 1) + 1(k - 1) = 0
(k - 1)(2k + 1) = 0
So, either
k - 1 = 0
k = 1
Or
2k + 1 = 0
2k = -1
k = -1/2
Therefore, the value of k = 1, -1/2
APPEARS IN
RELATED QUESTIONS
If -5 is a root of the quadratic equation 2x2 + px – 15 = 0 and the quadratic equation p(x2 + x)k = 0 has equal roots, find the value of k.
If x=−`1/2`, is a solution of the quadratic equation 3x2+2kx−3=0, find the value of k
Find the values of k for which the quadratic equation (3k + 1) x2 + 2(k + 1) x + 1 = 0 has equal roots. Also, find the roots.
Find the values of k for the following quadratic equation, so that they have two equal roots.
2x2 + kx + 3 = 0
Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.
In the following determine the set of values of k for which the given quadratic equation has real roots:
2x2 + kx + 2 = 0
Write the value of k for which the quadratic equation x2 − kx + 4 = 0 has equal roots.
Find the value of the discriminant in the following quadratic equation :
`4 sqrt 3 "x"^2 + 5"x" - 2 sqrt 3 = 0`
Determine the nature of the roots of the following quadratic equation :
2x2 + 5x - 6 = 0
Solve the following quadratic equation using formula method only
x2 - 4x - 1 = 0
Solve the following quadratic equation using formula method only
25x2 + 30x + 7 = 0
Solve the following quadratic equation using formula method only
16x2 - 24x = 1
Find the value of k for which the roots of the equation 3x2 - 10x + k = 0 are reciprocal of each other.
(3x - 5)(2x + 7) = 0
Find the discriminant of the following equations and hence find the nature of roots: 3x2 – 5x – 2 = 0
Find the values of k for which each of the following quadratic equation has equal roots: 9x2 + kx + 1 = 0 Also, find the roots for those values of k in each case.
If the roots of the given quadratic equation are real and equal, then find the value of ‘m’.
(m – 12)x2 + 2(m – 12)x + 2 = 0
If α, β are roots of the equation x2 + px – q = 0 and γ, δ are roots of x2 + px + r = 0, then the value of (α – y)(α – δ) is ______.
The roots of the quadratic equation px2 – qx + r = 0 are real and equal if ______.
Which of the following equations has two real and distinct roots?
