Question

find the discriminant of the equations 2x^2-5x+3=8 and hence write the narure of roots​

Answer 1

${Solution}$

Given to find the discriminant and nature of Quadratic equation :-

2x² - 5x + 3 = 8

SOLUTION:-

The discriminant of a Quadratic equation is given by b²-4ac.Discriminant of Quadratic equation is denoted by D.To find the nature of roots there are some cases

• D = 0 Roots are real and equal
• D> 0 Roots real and distinct
• D<0 Roots are complex and conjugate to each other

2x² - 5x + 3 = 8

2x² - 5x + 3 -8 = 0

2x² -5x -5 = 0

Comparing with ax² + bx + c

• a = 2
• b = -5
• c = -5

D = b²-4ac

D = (-5)² -4 (2) (-5)

D = 25 + 40

D = 65

Hence Discriminant of Quadratic equation is 65

Hence D>0 So, roots are real and distinct

Nature of roots :- Real and Distinct

Know more :-

• A Quadratic equatuion is one type of equation whose degree is 2
• A Quadratic equation has 2 roots
• We can find the two roots by various methods like complete squaring, factorisation etc

Answer 2

Answer:

Given :-

• 2x² - 5x + 3 = 8

To Find :-

• What is the discriminate of the equation.
• What is the nature of the roots.

Solution :-

Given equation :

$\bigstar \: \sf\bold{\green{2x^2 - 5x + 3 =\: 8}}$

Now, by solving this equation we get,

$\implies \sf 2x^2 - 5x + 3 =\: 8$

$\implies \sf 2x^2 - 5x + 3 - 8 =\: 0$

$\implies \sf 2x^2 - 5x - 5 =\: 0$

$\implies \sf\bold{\purple{2x^2 - 5x - 5 =\: 0}}$

Hence, we get the equation :

$\dashrightarrow \sf\bold{\green{2x^2 - 5x - 5 = 0}}$

where,

• a = 2
• b = - 5
• c = - 5

Now, as we know that :

$\clubsuit$ Discriminate Formula :

$\longmapsto \sf\boxed{\bold{\pink{Discriminate\: (D) =\: b^2 - 4ac}}}$

According to the question by using the formula we get,

$\implies \sf Discriminate\: (D) =\: (- 5)^2 - 4(2)(- 5)$

$\implies \sf Discriminate\: (D) =\: (- 5) \times (- 5) - 8(- 5)$

$\implies \sf Discriminate\: (D) =\: 25 - (- 40)$

$\implies \sf Discriminate\: (D) =\: 25 + 40$

$\implies \sf\bold{\red{Discriminate\: (D) =\: 65\: >\: 0}}$

$\therefore$ The discriminate of the equation is 65.

$\therefore$ The nature of the roots is real and distinct.

$\rule{150}{2}$

EXTRA INFORMATION :

❒ The general form of the type of equation is ax² + bx + c = 0.

[ If a = 0 then the equation becomes to a linear equation.

If b = 0 then the roots of the equation becomes equal but opposite in sign.

If c = 0 then one of the roots is zero.]

❒ When b² - 4ac = 0 then the roots are real & equal.

❒ When b² - 4ac > 0 then the roots are real and distinct.

❒ When b² - 4ac < 0 then there will be no roots.