Question

find nature of roots of quadratic equation 3x^2+4x-8=0​

Answer 1

$\huge{\mathfrak{\underline{\underline{\red{Answer :-}}}}}$

Real and two distinct roots

$\large{\mathrm{\underline{\gray{Given :-}}}}$

Equation :- 3x² + 4x - 8 = 0

a = 3

b = 4

c = -8

$\large{\mathrm{\underline{\gray{To \: Find :-}}}}$

Nature of roots

$\large{\mathrm{\underline{\gray{Solution :-}}}}$

[Quadratic formula]

$\huge{\boxed{\boxed{\bf{D \: = \: b^{2} \: - \: 4ac}}}}$

___________________[Put Values]

D = (3)² - 4(3)(-8)

D = 9 - (-96)

D = 9 + 96

D = 105

✯ We know that when D is greater than zero. Then, equation real and Distinct roots

$\large{\boxed{\boxed{Real \: and \: equal \: roots}}}$

$\large{\mathrm{\underline{\gray{Extra \: information :-}}}}$

√D = √105

Now to find zeroes we have a formula :-

$\huge{\bf{\boxed{\boxed{x \: = \: {\frac{-b \: ± \: {\sqrt{D}}}{2a}}}}}}$

______________[Put values]

x = (- 4 ± √105)/2(3)

x = (-4 ± √185)/6

_________________________

✯ Case 1

x = (-4 + √105)/6

________________________

✯ Case 2

x = (-4 - √105)/6

_________________________

$\large{\bf{\boxed{\boxed{x \: = \frac{ - 4 + \sqrt{105} }{6} }}}}$

Or

$\large{\boxed{\boxed{x \: = \: \frac{ - 4 - \sqrt{105} }{6} }}}$