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Nature of Roots

In the follow...

Question

In the following, determine the set values of k for which the given quadratic equation has real roots:

(i) $2 x^{2} + 3 x + k = 0$
(ii) $2 x^{2} + k x + 3 = 0$
(iii) $2 x^{2} - 5 x - k = 0$
(iv) $k x^{2} + 6 x + 1 = 0$
(v) $x^{2} - k x + 9 = 0$
(vi) $2 x^{2} + k x + 2 = 0$
(vii) $3 x^{2} + 2 x + k = 0$
(viii) $4 x^{2} - 3 k x + 1 = 0$
(ix) $2 x^{2} + k x - 4 = 0$

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Solution

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

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real roots, if

Therefore, the value of

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

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real roots, if

Therefore, the value of

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

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real roots, if

Therefore, the value of

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

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real roots, if

Therefore, the value of

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

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real roots, if

Therefore, the value of

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

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real roots, if

Therefore, the value of

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

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real roots, if

Therefore, the value of

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

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real roots, if

Therefore, the value of

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

Then find the value of k.

Here,

As we know that

Putting the value of

The given equation will have real roots, if

Since left hand side is always positive. So

Therefore, the value of

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Similar questions

Q. Find the values of k for which the following equations have real roots

(i)

x^{2} - 4 k x + k = 0

k x (x - 2 \sqrt{5}) + 10 = 0

k x (x - 3) + 9 = 0

4 x^{2} + k x + 3 = 0

Q. 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

k x^{2} - 2 \sqrt{5} x + 4 = 0

3 x^{2} - 5 x + 2 k = 0

4 x^{2} + k x + 9 = 0

2 k x^{2} - 40 x + 25 = 0

9 x^{2} - 24 x + k = 0

4 x^{2} - 3 k x + 1 = 0

x^{2} - 2 (5 + 2 k) x + 3 (7 + 10 k) = 0

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

k x^{2} + k x + 1 = - 4 x^{2} - x

(k + 1) x^{2} + 2 (k + 3) x + (k + 8) = 0

x^{2} - 2 k x + 7 k - 12 = 0

(k + 1) x^{2} - 2 (3 k + 1) x + 8 k + 1 = 0

(2 k + 1) x^{2} + 2 (k + 3) x + (k + 5) = 0

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

Q. In the following, determine whether the given quadratic equations have real roots and if so, find the roots:

(i) 16x² = 24x + 1
(ii) x² + x + 2 = 0
(iii)

\sqrt{3} x^{2} + 10 x - 8 \sqrt{3} = 0

(iv) 3x² − 2x + 2 = 0
(v)

2 x^{2} - 2 \sqrt{6} x + 3 = 0

3 a^{2} x^{2} + 8 a b x + 4 b^{2} = 0, a \neq 0

3 x^{2} + 2 \sqrt{5} x - 5 = 0

(viii) x² − 2x + 1 = 0
(ix)

2 x^{2} + 5 \sqrt{3} x + 6 = 0

\sqrt{2} x^{2} + 7 x + 5 \sqrt{2} = 0

2 x^{2} - 2 \sqrt{2} x + 1 = 0

(xii) 3x² − 5x + 2 = 0

Q. The set of values of k for which the given quadratic equation has real roots

2 x^{2} + k x + 2 = 0

\leq

Q. Find the roots of the quadratic equations by using the quadratic formula in the following equation:
(i)

2 x^{2} - 3 x - 5 = 0

5 x^{2} + 13 x + 8 = 0

- 3 x^{2} + 5 x + 12 = 0

- x^{2} + 7 x - 10 = 0

x^{2} + 2 \sqrt{2} x - 6 = 0

x^{2} - 3 \sqrt{5} x + 10 = 0

\frac{1}{2} x^{2} - \sqrt{11} x + 1 = 0

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