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
Find the values of k for the following quadratic equation, so that they have two equal roots.
kx (x - 2) + 6 = 0
Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
Find the values of k for which the roots are real and equal in each of the following equation:
x2 - 4kx + k = 0
Solve for x :
x2 + 5x − (a2 + a − 6) = 0
Find the value(s) of k so that the quadratic equation 3x2 − 2kx + 12 = 0 has equal roots ?
Determine the nature of the roots of the following quadratic equation :
4x2 - 8x + 5 = 0
Solve the following quadratic equation using formula method only
3a2x2 +8abx + 4b2 = 0, a ≠ 0
`10x -(1)/x` = 3
Without solving the following quadratic equation, find the value of ‘p’ for which the given equation has real and equal roots:
x² + (p – 3) x + p = 0
Without actually determining the roots comment upon the nature of the roots of each of the following equations:
x2 - 5x + 7 = 0
Determine whether the given quadratic equations have equal roots and if so, find the roots:
3x2 - 6x + 5 = 0
If the ratio of the roots of the equation
lx2 + nx + n = 0 is p: q, Prove that
`sqrt(p/q) + sqrt(q/p) + sqrt(n/l) = 0.`
Find the discriminant of the following equations and hence find the nature of roots: 7x2 + 8x + 2 = 0
If a = 1, b = 4, c = – 5, then find the value of b2 – 4ac
If x2 (a2 + b2) + 2x (ac + bd) + c2 +d2 = 0 has no real roots, then:
Which constant must be added and subtracted to solve the quadratic equation `9x^2 + 3/4x - sqrt(2) = 0` by the method of completing the square?
If the coefficient of x2 and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.
The roots of quadratic equation x2 – 1 = 0 are ______.
The roots of quadratic equation x(x + 8) + 12 = 0 are ______.
If the roots of x2 – px + 4 = 0 are equal, the value (values) of p is ______.
