Advertisements
Advertisements
Question
For what value of k, (4 - k)x2 + (2k + 4)x + (8k + 1) = 0, is a perfect square.
Advertisements
Solution
The given quadric equation is (4 - k)x2 + (2k + 4)x + (8k + 1) = 0, and roots are real and equal
Then find the value of k.
Here, a = (4 - k), b = (2k + 4) and c = (8k + 1)
As we know that D = b2 - 4ac
Putting the value of a = (4 - k), b = (2k + 4) and c = (8k + 1)
= (2k + 4)2 - 4 x (4 - k) x (8k + 1)
= 4k2 + 16k - 16 - 4(4 + 31k - 8k2)
= 4k2 + 16k - 16 - 16 - 124k + 32k2
= 36k2 - 108k + 0
= 36k2 - 108k
The given equation will have real and equal roots, if D = 0
Thus,
36k2 - 108k = 0
18k(2k - 6) = 0
k(2k - 6) = 0
Now factorizing of the above equation
k(2k - 6) = 0
So, either
k = 0
or
2k - 6 = 0
2k = 6
k = 6/2 = 3
Therefore, the value of k = 0, 3.
APPEARS IN
RELATED QUESTIONS
Find that value of p for which the quadratic equation (p + 1)x2 − 6(p + 1)x + 3(p + 9) = 0, p ≠ − 1 has equal roots. Hence find the roots of the equation.
Determine the nature of the roots of the following quadratic equation:
(x - 2a)(x - 2b) = 4ab
Find the value of k for which the roots are real and equal in the following equation:
3x2 − 5x + 2k = 0
In the following determine the set of values of k for which the given quadratic equation has real roots:
2x2 + kx - 4 = 0
What is the nature of roots of the quadratic equation 4x2 − 12x − 9 = 0?
Find the value of the discriminant in the following quadratic equation:
2x2 - 3x + 1 = O
`sqrt(3)x^2 + 11x + 6sqrt(3)` = 0
In each of the following determine the; value of k for which the given value is a solution of the equation:
3x2 + 2kx - 3 = 0; x = `-(1)/(2)`
Determine whether the given quadratic equations have equal roots and if so, find the roots:
`(4)/(3)x^2 - 2x + (3)/(4) = 0`
If a is a root of the equation x2 – (a + b)x + k = 0, find the value of k.
If the root of the given quadratic equation are real and equal, then find the value of ‘k’ X2 + 2X + k = 0.
The roots of the quadratic equation `2"x"^2 - 2sqrt2"x" + 1 = 0` are:
α and β are the roots of 4x2 + 3x + 7 = 0, then the value of `1/α + 1/β` is:
The roots of the quadratic equation `1/("a" + "b" + "x") = 1/"a" + 1/"b" + 1/"x"`, a + b ≠ 0 is:
State whether the following quadratic equation have two distinct real roots. Justify your answer.
x2 – 3x + 4 = 0
State whether the following quadratic equation have two distinct real roots. Justify your answer.
`2x^2 - 6x + 9/2 = 0`
Every quadratic equation has at least one real root.
State whether the following quadratic equation have two distinct real roots. Justify your answer.
`(x - sqrt(2))^2 - 2(x + 1) = 0`
Find the value of k for which the roots of the quadratic equation 5x2 – 10x + k = 0 are real and equal.
The roots of the quadratic equation x2 – 6x – 7 = 0 are ______.
