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

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Solution

(i) $D = (- 3)^{2} - 4 \times 2 \times 5 = 9 - 40 = - 31$

As D is negative, roots are not real.

(ii) $D = (- 6)^{2} - 4 \times 3 \times 3 = 36 - 36 = 0$

As D is zero, roots are real and equal.

(iii) $D = {(\frac{- 2}{3})}^{2} - 4 \times \frac{3}{5} \times 1 = \frac{4}{9} - \frac{12}{5} < 0$

As D is negative, roots are not real.

(iv) $D = (- 4 \sqrt{3})^{2} - 4 \times 3 \times 4 = 48 - 48 = 0$

As D is zero, roots are real and equal.

(v) $D = (- 2 \sqrt{6})^{2} - 4 \times 3 \times 2 = 24 - 24 = 0$

As D is zero, roots are real and equal.

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Similar questions

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

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

Q. Determine the nature of the roots of the following quadratic equations:

(i) 2x² − 3x + 5 = 0
(ii) 2x² − 6x + 3 = 0
(iii)

\frac{3}{5} x^{2} - \frac{2}{3} x + 1 = 0

(iv)

3 x^{2} - 4 \sqrt{3} x + 4 = 0

(v)

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

Which of the following are quadratic equations?
$(i) x^{2} + 6 x - 4 = 0$
( $i i) \sqrt{3 x^{2}} - 2 x + \frac{1}{2} = 0$
$(i i i) x^{2} + \frac{1}{x^{2}} = 5$
$(i v) x - \frac{3}{x} = x^{2}$
$(v) 2 x^{2} - \sqrt{3 x} + 9 = 0$
$(v i) x^{2} - 2 x - \sqrt{x} - 5 = 0$
$(v i i) 3 x^{2} - 5 x + 9 = x^{2} - 7 x + 3$
$(v i i i) x + \frac{1}{x} = 1$
$(i x) x^{2} - 3 x = 0$
$(x) {(x + \frac{1}{x})}^{2} = 3 (x + \frac{1}{x}) + 4$
$(x i) (2 x + 1) (3 x + 2) = 6 (x - 1) (x - 2)$
$(x i i) x + \frac{1}{x} = x^{2}, x \neq 0$
$(x i i i) 16 x^{2} - 3 = (2 x + 5) (5 x - 3)$
$(x i v) (x + 2)^{3} = x^{3} - 4$
$(x v) x (x + 1) + 8 = (x \div 2) (x - 2)$

Q. The roots of the quadratic equation

2 (3 x + 5) (2 x - 4) = 0

are

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Determine the nature of the roots of the following quadratic equations: (i)2x2−3x+5=0(ii)3x2−6x+3=0(iii)35x2−23x+1=0(iv)3x2−4√3x+4=0(v)3x2−2√6x+2=0

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