Advertisements
Advertisements
प्रश्न
Find the values of k for which the roots are real and equal in each of the following equation:
x2 - 2(5 + 2k)x + 3(7 + 10k) = 0
Advertisements
उत्तर
The given quadric equation is x2 - 2(5 + 2k)x + 3(7 + 10k) = 0, and roots are real and equal
Then find the value of k.
Here, a = 1, b = -2(5 + 2k) and c = 3(7 + 10k)
As we know that D = b2 - 4ac
Putting the value of a = 1, b = -2(5 + 2k) and c = 3(7 + 10k)
= (-2(5 + 2k))2 - 4 x (1) x 3(7 + 10k)
= 4(25 + 20k + 4k2) - 12(7 + 10k)
= 100 + 80k + 16k2 - 84 - 120k
= 16 - 40k + 16k2
The given equation will have real and equal roots, if D = 0
Thus,
16 - 40k + 16k2 = 0
8(2k2 - 5k + 2) = 0
2k2 - 5k + 2 = 0
Now factorizing of the above equation
2k2 - 5k + 2 = 0
2k2 - 4k - k + 2 = 0
2k(k - 2) - 1(k - 2) = 0
(k - 2)(2k - 1) = 0
So, either
k - 2 = 0
k = 2
Or
2k - 1 = 0
2k = 1
k = 1/2
Therefore, the value of k = 2, 1/2.
APPEARS IN
संबंधित प्रश्न
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.
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.
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.
Is it possible to design a rectangular park of perimeter 80 and area 400 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:
4x2 - 3kx + 1 = 0
Show that the equation 2(a2 + b2)x2 + 2(a + b)x + 1 = 0 has no real roots, when a ≠ b.
The equation `3x^2 – 12x + (n – 5) = 0` has equal roots. Find the value of n.
Solve the following quadratic equation using formula method only
5x2 - 19x + 17 = 0
Solve the following quadratic equation using formula method only
`sqrt 3 "x"^2 + 10 "x" - 8 sqrt 3 = 0`
Solve the following quadratic equation using formula method only
`"x"^2 + 1/2 "x" - 1 = 0`
For what value of k, the roots of the equation x2 + 4x + k = 0 are real?
If x = 2 and x = 3 are roots of the equation 3x² – 2kx + 2m = 0. Find the values of k and m.
Find the value of k for which the given equation has real roots:
kx2 - 6x - 2 = 0
Without actually determining the roots comment upon the nature of the roots of each of the following equations:
9a2b2x2 - 48abc + 64c2d2 = 0, a ≠ 0, b ≠ 0
Solve the following by reducing them to quadratic equations:
x4 - 26x2 + 25 = 0
In each of the following, determine whether the given numbers are roots of the given equations or not; x2 – x + 1 = 0; 1, – 1
The roots of the quadratic equation `2"x"^2 - 2sqrt2"x" + 1 = 0` are:
If the one root of the equation 4x2 – 2x + p – 4 = 0 be the reciprocal of the other. The value of p is:
If the coefficient of x2 and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.
The sum of all integral values of k(k ≠ 0) for which the equation `2/(x - 1), 1/(x - 2) = 2/k` in x has no real roots, is ______.
