Question

find the nature of the roots of 3x^2+5x+2=0

Answer 1

${3x}^{2} + 5x + 2 = 0 \\ {3x}^{2} + (3 + 2)x + 2 = 0 \\ 3 {x}^{2} + 3x + 2x + 2 = 0 \\ 3x(x + 1) + 2(x + 1) = 0 \\ (3x + 2)(x + 1) = 0 \\ \\ now \\ \\ 3x + 2 = 0 \\ 3x = - 2 \\ x = \frac{ - 2}{3} \\ \\ x + 1 = 0 \\ x = - 1$

Answer 2

Answer:

Nature of roots of a quadratic equation can be calculated by calculating the value of discriminant.

Discriminant = b² - 4ac

This is denoted by the letter D

Case 1 : D > 0

If D is greater than 0, then we get two real and distinct roots.

Case 2: D = 0

If D is equal to 0, then we get two roots which are equal and same.

Case 3: D < 0

If D is less than 0, then we get roots which are imaginary or unreal.

According to this question,

Equation: 3x² + 5x + 2 = 0

Here, a = 3, b = 5, c = 2

Calculating the discriminant we get,

⇒ D = ( 5 )² - 4 ( 3 ) ( 2 )

⇒ D = 25 - 4 ( 6 )

⇒ D = 25 - 24 = 1

Since D is greater than 0 in this case, we get two real and distinct roots.

Hence solved !!