Find the nature of the roots of the following quadratic equation:
1. $3 x^{2} + 7 x + 2 = 0$
2. ${(x + 3)}^{2} + 4 x + 7 = 0$
3. $2 x^{2} - 8 x + 34 = 0$
4. $x^{2} + 2 \sqrt{3 x} - 1 = 0$

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Solution

A. $3 x^{2} + 7 x + 2 = 0$
$△ = b^{2} - 4 a c = 7^{2} - 4 (2) (3) = 49 - 24 = 25 > 0$
$∴$ Roots are real and positive
B. ${(x + 3)}^{2} + 4 x + 7 = 0$
$x^{2} + 9 + 6 x + 4 x + 7 = 0$
$x^{2} + 10 x + 16 = 0$
$△ = b^{2} - 4 a c = 100 - 4 (16) = 36 = 6^{2} > 0$
$∴$ Roots are real and positive
C. $2 x^{2} - 8 x + 34 = 0$
$△ = 8^{2} - 4 (2) (34) < 0$
$∴$ Roots are imaginary
D. $x^{2} + 2 \sqrt{3} x - 1 = 0$
$△ = {(2 \sqrt{3})}^{2} + 4 (1) = 12 + 4 = 16 > 0$
Roots are real and positive.

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