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Standard X

Mathematics

Nature of Roots

Find the valu...

Question

Find the values of k for which th efollowing equations have real roots.
$(i) 2 x^{2} + k x + 3 = 0 (i i) k x (x - 2) + 6 = 0 (i i i) x^{2} - 4 k x + k = 0 (i v) k x (x - 2 \sqrt{5}) + 10 = 0 (v) k x (x - 3) + 9 = 0 (v i) 4 x^{2} + k x + 3 = 0$

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Solution

$(i) k = \pm 2 \sqrt{6} (i i) k = 6 (i i i) k = 0, \frac{1}{4} (i v) k = 2 (v) k = 4 (v i) k = \pm 4 \sqrt{3}$

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Similar questions

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

(i)

(ii)

(iii)

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

(iv)

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

(v)

k x (x - 3) + 9 = 0

(vi)

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

In the following, determine the set of values of k for which the given quadratic equation has real roots:
$(i) 2 x^{2} + 3 x + k = 0 (i i) 2 x^{2} + x + k = 0 (i i i) 2 x^{2} - 5 x - k = 0 (i v) k x^{2} + 6 x + 1 = 0 (v) 3 x^{2} + 2 x + k = 0$

Q. 1. Find the values of

k

for which the following equations have real and equal roots:

\begin{matrix} (i) x^{2} - 2 (k + 1) x + k^{2} = 0 (ii) k^{2} x^{2} - 2 (2 k - 1) x + 4 = 0 (iii) (k + 1) x^{2} - 2 (k - 1) x + 1 = 0 (iv) x^{2} + k (2 x + k - 1) + 2 = 0 \end{matrix}

2. Find the values of

k

for which the following equations have real roots:

\begin{matrix} (i) 2 x^{2} + k x + 3 = 0 (ii) x^{2} - 4 k x + k = 0 (iii) k x (x - 3) + 9 = 0 \end{matrix}

\begin{matrix} (iv) k x (x - 2) + 6 = 0 (v) k x (x - 2 \sqrt{5}) + 10 = 0 (vi) 4 x^{2} + k x + 3 = 0 \end{matrix}

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

(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)

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

(xix)

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

(xx)

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

(xxi)

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

(xxii)

k x (x - 2) + 6 = 0

(xxiii)

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

(xxiv)

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

(xxv)

p x (x - 3) + 9 = 0

(xxvi)

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

Q. Find the values of

k

for which the following equations have real roots.

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

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Find the values of k for which th efollowing equations have real roots. (i)2x2+kx+3=0(ii)kx(x−2)+6=0(iii)x2−4kx+k=0(iv)kx(x−2√5)+10=0(v)kx(x−3)+9=0(vi)4x2+kx+3=0

(i)k=±2√6(ii)k=6(iii)k=0,14(iv)k=2(v)k=4(vi)k=±4√3

Find the values of k for which th efollowing equations have real roots.
$(i) 2 x^{2} + k x + 3 = 0 (i i) k x (x - 2) + 6 = 0 (i i i) x^{2} - 4 k x + k = 0 (i v) k x (x - 2 \sqrt{5}) + 10 = 0 (v) k x (x - 3) + 9 = 0 (v i) 4 x^{2} + k x + 3 = 0$

$(i) k = \pm 2 \sqrt{6} (i i) k = 6 (i i i) k = 0, \frac{1}{4} (i v) k = 2 (v) k = 4 (v i) k = \pm 4 \sqrt{3}$