Advertisements
Advertisements
Question
In the following determine the set of values of k for which the given quadratic equation has real roots:
2x2 − 5x − k = 0
Advertisements
Solution
The given quadric equation is 2x2 − 5x − k = 0, and roots are real
Then find the value of k.
Here,
a = 2
b = −5
c = k
As we know that D = b2 − 4ac
Putting the value of a = 2, b = −5 and c = k
= (−5)2 − 4 × (2) × (−k)
= 25 + 8k
The given equation will have real roots, if D ≥ 0
25 + 8k ≥ 0
8k ≥ −25
k ≥ `−25/8`
Therefore, the value of k ≥ `−25/8`
APPEARS IN
RELATED QUESTIONS
For what value of m, are the roots of the equation (3m + 1)x2 + (11 + m) x + 9 = 0 equal?
Find the nature of the roots of the following quadratic equation. If the real roots exist, find them:
2x2 – 3x + 5 = 0
Find the nature of the roots of the following quadratic equation. If the real roots exist, find them:
2x2 – 6x + 3 = 0
The 4th term of an A.P. is 22, and the 15th term is 66. Find the first term and the common difference. Hence, find the sum of the series to 8 terms.
Find the values of k for which the roots are real and equal in each of the following equation:
(k + 1)x2 - 2(3k + 1)x + 8k + 1 = 0
Prove that both the roots of the equation (x – a)(x – b) + (x – b)(x – c) + (x – c)(x – a) = 0 are real but they are equal only when a = b = c.
Determine the nature of the roots of the following quadratic equation :
(x - 1)(2x - 7) = 0
Find the value of m for which the equation (m + 4)x2 + (m + 1)x + 1 = 0 has real and equal roots.
Find if x = – 1 is a root of the equation 2x² – 3x + 1 = 0.
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
Form the quadratic equation whose roots are:
`sqrt(3) and 3sqrt(3)`
Determine whether the given quadratic equations have equal roots and if so, find the roots:
x2 + 5x + 5 = 0
Determine whether the given quadratic equations have equal roots and if so, find the roots:
`(4)/(3)x^2 - 2x + (3)/(4) = 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
If `(2)/(3)` and – 3 are the roots of the equation px2+ 7x + q = 0, find the values of p and q.
Find the discriminant of the following equations and hence find the nature of roots: 16x2 - 40x + 25 = 0
Discuss the nature of the roots of the following quadratic equations : `x^2 - (1)/(2)x + 4` = 0
Find the least positive value of k for which the equation x2 + kx + 4 = 0 has real roots.
Find the value(s) of k for which each of the following quadratic equation has equal roots: 3kx2 = 4(kx – 1)
If a = 1, b = 4, c = –5, then find the value of b2 – 4ac.
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
The quadratic equation whose one rational root is `3 + sqrt2` is
If the equation x2 – (2 + m)x + (–m2 – 4m – 4) = 0 has coincident roots, then:
State whether the following quadratic equation have two distinct real roots. Justify your answer.
(x + 4)2 – 8x = 0
State whether the following quadratic equation have two distinct real roots. Justify your answer.
x(1 – x) – 2 = 0
Solve the equation: 3x2 – 8x – 1 = 0 for x.
The probability of selecting integers a ∈ [–5, 30] such that x2 + 2(a + 4)x – 5a + 64 > 0, for all x ∈ R, is ______.
If α and β are the distinct roots of the equation `x^2 + (3)^(1/4)x + 3^(1/2)` = 0, then the value of α96(α12 – 1) + β96(β12 – 1) is equal to ______.
The nature of roots of the quadratic equation 9x2 – 6x – 2 = 0 is ______.
When the discriminant satisfies \( D > 0 \), what is the nature of the roots of the quadratic equation?
