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Existence of Limit

Find the valu...

Question

Find the values of k for which the roots are real and equal in each of the following equations:

(i) $k x^{2} + 4 x + 1 = 0$
(ii) $k x^{2} - 2 \sqrt{5} x + 4 = 0$
(iii) $3 x^{2} - 5 x + 2 k = 0$
(iv) $4 x^{2} + k x + 9 = 0$
(v) $2 k x^{2} - 40 x + 25 = 0$
(vi) $9 x^{2} - 24 x + k = 0$
(vii) $4 x^{2} - 3 k x + 1 = 0$
(viii) $x^{2} - 2 (5 + 2 k) x + 3 (7 + 10 k) = 0$
(ix) $(3 k + 1) x^{2} + 2 (k + 1) x + k = 0$
(x) $k x^{2} + k x + 1 = - 4 x^{2} - x$
(xi) $(k + 1) x^{2} + 2 (k + 3) x + (k + 8) = 0$
(xii) $x^{2} - 2 k x + 7 k - 12 = 0$
(xiii) $(k + 1) x^{2} - 2 (3 k + 1) x + 8 k + 1 = 0$
(xiv) $(2 k + 1) x^{2} + 2 (k + 3) x + (k + 5) = 0$
(xvii) $4 x^{2} - 2 (k + 1) x + (k + 4) = 0$
(xviii) $4 x^{2} - 2 (k + 1) x + (k + 1) = 0$

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Solution

(i) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if

Thus,

Therefore, the value of

(ii) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Thus,

Therefore, the value of

(iii) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Thus,

Therefore, the value of

(iv) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Thus,

Therefore, the value of

(v) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Thus,

Therefore, the value of

(vi) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Thus,

Therefore, the value of

(vii) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Thus,

Therefore, the value of

(viii) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Thus,

Now factorizing of the above equation

So, either

Therefore, the value of

(ix) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Thus,

Now factorizing of the above equation

So, either

Therefore, the value of

(x) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

So,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Thus,

Now factorizing of the above equation

So, either

Therefore, the value of

(xi) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Therefore, the value of

(xii) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Now factorizing of the above equation

So, either

Therefore, the value of

(xiii) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Now factorizing of the above equation

So, either

Therefore, the value of

(xiv) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

So,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Thus,

Now factorizing of the above equation

So, either

Therefore, the value of

(xv) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Now factorizing of the above equation

So, either

Therefore, the value of

(xvi) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Now factorizing the above equation

So, either

Therefore, the value of

(xvii) The given quadric equation is , and roots are real and equal

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real and equal roots, if D = 0

Now factorizing of the above equation

So, either

Therefore, the value of

(xviii) The given equation is $4 x^{2} - 2 (k + 1) x + (k + 1) = 0$ where a = 4, b = -2(k+1), c = (k+1)
As we know that
Putting the value of a = 4, b = -2(k+1), c = (k+1)
${\{- 2 (k + 1)\}}^{2} - 4 \times 4 \times (K + 1) 4 (K + 1)^{2} - 16 (K + 1) (K + 1) \{4 (K + 1) - 16\} (K + 1) (4 K - 12) 4 (K + 1) (K - 3)$
For real and equal roots D = 0

$4 (K + 1) (K - 3) = 0 K = - 1 or k = 3$

Therefore, the value of

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Similar questions

Find the values of k for which the roots are real and equal in each of the following equations.
$(i) k x^{2} + 4 x + 1 = 0 (i i) k x^{2} - 2 \sqrt{5} x + 4 = 0 (i i i) 3 x^{2} - 5 x + 2 k = 0 (i v) 4 x^{2} + k x + 9 = 0 (v) 2 k x^{2} - 40 x + 25 = 0 (v i) 9 x^{2} - 24 x + k = 0 (v i i) 4 x^{2} - 2 k x + 1 = 0 (v i i i) x^{2} - 2 (5 + 2 k) x + 3 (7 + 10 k) = 0 (i x) (3 k + 1) x^{2} + 2 (k + 1) x + k = 0 (x) k x^{2} + k x + 1 = - 4 x^{2} - x (x i) (k + 1) x^{2} + 2 (k + 3) x (k + 8) = 0 (x i i) x^{2} - 2 k x + 7 k - 12 = 0 (x i i i) (k + 1) x^{2} - 2 (3 k + 1) x + 8 k + 1 = 0 (x i v) 5 x^{2} - 4 x + 2 + k (4 x^{2} 1 - 2 x - 1) = 0 (x v) (4 - k) x^{2} + (2 k + 4) x + (8 k + 1) = 0 (x v i) (2 k + 1) x^{2} + 2 (k + 3) x + (k + 5) = 0 (x v i i) 4 x^{2} - 2 (k + 1) x + (k + 4) = 0 (x v i i i) 4 x^{2} - 2 (k + 1) x + (k + 1) = 0$

Q. Find the values of k for which the following equations have real and equal roots:
(i)

x^{2} - 2 (k + 1) x + k^{2} = 0

k^{2} x^{2} - 2 (2 k - 1) x + 4 = 0

(k + 1) x^{2} - 2 (k - 1) x + 1 = 0

x^{2} + k (2 x + k - 1) + 2 = 0

(k + 1) x^{2} - 2 (k - 1) x + 1 = 0

Q. Find the value of

K

4 x^{2} - 2 (K + 1) x + (K + 4) = 0

has real and equal roots

k

, so that the quadratic equation

(k + 1) x^{2} - 2 (k + 1) x + 1 = 0

has equal roots.

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