Question

Determine the nature of roots for each of the
quadratic equation
1) 3 x 2-5x+7=0​

Answer 1

EXPLANATION.

Quadratic equation.

⇒ 3x² - 5x + 7 = 0.

As we know that,

D = Discriminant  Or  b² - 4ac.

⇒ D = (-5)² - 4(3)(7).

⇒ D = 25 - 84.

⇒ D = -59.

⇒ D < 0 Roots are imaginary.

                                                                                                                     

MORE INFORMATION.

Nature of the roots of the quadratic expression.

(1) = Real and different, if b² - 4ac > 0.

(2) = Rational and different, if b² - 4ac is a perfect square.

(3) = Real and equal, if b² - 4ac = 0.

(4) = If D < 0 Roots are imaginary and unequal or complex conjugate.

Answer 2

Given Question :-

• Determine the nature of roots for each of the quadratic equation : 3 x² - 5x + 7 = 0

Given :-

• A quadratic equation : 3 x² - 5x + 7 = 0

To Find :-

• Nature of roots of 3 x² - 5x + 7 = 0

Concept Used :-

Concept Used :- Nature of roots

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots depend upon Discriminant.

• If Discriminant, D > 0, then equation has real and unequal roots.

• If Discriminant, D = 0, then equation has real and equal roots.

• If Discriminant, D < 0, then equation has no real roots or imaginary roots or complex roots.

where,

• Discriminant, D = b² - 4ac

Let's do it now!!

$\large\underline\purple{\bold{Solution :- }}$

Given quadratic equation is

• 3 x² - 5x + 7 = 0

On comparing with ax² + bx + c = 0

we get

• a = 3

• b = - 5

• c = 7

Now,

• Discriminant, D of the equation is given by

$\tt \longmapsto\:D = {b}^{2} - 4ac$

$\tt \longmapsto\:D = {( - 5)}^{2} - 4 \times 3 \times 7$

$\tt \longmapsto\:D = 25 - 84$

$\tt \longmapsto\:D = - 59$

$\rm :\implies\: \: \boxed{ \green{ \bf \:D < 0 }}$

Hence,

• Quadratic equation has no real root