In the frollowing determine whetehr the given quadratic equations have real roots i 16x2 - 24x-1- ii x2-x-2 - 0- iii-3x2- 10x - 8 -3 - 0- iv3x2 - 2x-2 - 0- v 2x2-2 -3x-3 - 0- vi 3a2x2-8abx -4b2 - 0- a - 0- vii 3x2-1 -3x - 5 - 0- viiix2 - 2x-1 - 0 - ix2x2-5 -3x-6 - 0- x -2x2-7x-5 -2 - 0- xi2x2 - 2 -2x -1 - 0 - xii3x2-5x-2 - 0

(i)3 ±√(10)/4 (ii)not real (iii) 4 √(3), 2/√(3) (iv) Not real (v) Real and equal √(3/2) (vi) 2b/a , 2b/3a (vii) √(5)/3, 5 (viii)1 (ix) 2 √(3), √(3)/2 ( ...

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Byju's Answer

Standard X

Mathematics

Nature of Roots

In the frollo...

Question

In the frollowing determine whetehr the given quadratic equations have real roots
$(i) 16 x^{2} = 24 x + 1 (i i) x^{2} + x + 2 = 0 (i i i) \sqrt{3} x^{2} + 10 x - 8 \sqrt{3} = 0 (i v) 3 x^{2} - 2 x + 2 = 0 (v) 2 x^{2} - 2 \sqrt{3} x + 3 = 0 (v i) 3 a^{2} x^{2} + 8 a b x + 4 b^{2} = 0, a \neq 0 (v i i) 3 x^{2} + 1 \sqrt{3} x - 5 = 0 (v i i i) x^{2} - 2 x + 1 = 0 (i x) 2 x^{2} + 5 \sqrt{3} x + 6 = 0 (x) \sqrt{2} x^{2} + 7 x + 5 \sqrt{2} = 0 (x i) 2 x^{2} - 2 \sqrt{2} x + 1 = 0 (x i i) 3 x^{2} - 5 x + 2 = 0$

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Solution

$(i) \frac{3 \pm \sqrt{10}}{4} (i i) n o t r e a l (i i i) - 4 \sqrt{3}, \frac{2}{\sqrt{3}} (i v) N o t r e a l (v) R e a l a n d e q u a l \sqrt{\frac{3}{2}} (v i) \frac{- 2 b}{a}, - \frac{- 2 b}{3 a} (v i i) \frac{\sqrt{5}}{3}, - 5 (v i i i) 1 (i x) - 2 \sqrt{3}, \frac{- \sqrt{3}}{2} (x) - \sqrt{2}, - \frac{5}{\sqrt{2}} (x i) \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} (x i i) 1, \frac{2}{3}$

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Similar questions

Which of the following are quadratic equations?
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$(i v) x - \frac{3}{x} = x^{2}$
$(v) 2 x^{2} - \sqrt{3 x} + 9 = 0$
$(v i) x^{2} - 2 x - \sqrt{x} - 5 = 0$
$(v i i) 3 x^{2} - 5 x + 9 = x^{2} - 7 x + 3$
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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. Find whether the following equation have real roots. If real roots exist, find them
(i)

8 x^{2} + 2 x - 3 = 0

(ii)

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

(iii)

5 x^{2} - 2 x - 10 = 0

(iv)

\frac{1}{2 x - 3} + \frac{1}{x - 5} = 1, x \neq \frac{3}{2}, 5

(v)

x^{2} + 5 \sqrt{5} x - 70 = 0

Determine the nature of the roots of the following quadratic equations:
$(i) 2 x^{2} - 3 x + 5 = 0 (i i) 3 x^{2} - 6 x + 3 = 0 (i i i) \frac{3}{5} x^{2} - \frac{2}{3} x + 1 = 0 (i v) 3 x^{2} - 4 \sqrt{3} x + 4 = 0 (v) 3 x^{2} - 2 \sqrt{6} x + 2 = 0$

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

(ii)

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

(iii)

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

(iv)

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

(v)

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

(vi)

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

(vii)

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

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(i)3±√104(ii)notreal(iii)−4√3,2√3(iv)Notreal(v)Realandequal√32(vi)−2ba,−−2b3a(vii)√53,−5(viii)1(ix)−2√3,−√32(x)−√2,−5√2(xi)1√2,1√2(xii)1,23