Verified answer

[tex]\large\underline{\sf{Solution-}}[/tex]

Given quadratic equation is

[tex]\sf \: {2x}^{2} - 3x + 5 = 0 \\ \\ [/tex]

On comparing with ax² + bx + c = 0, we get

[tex]\boxed{\begin{aligned}& \qquad \: a=2 \qquad \:\\& \qquad \:b= - 3 \qquad \:\\& \qquad \:c=5 \qquad \:\end{aligned}} \\ \\ [/tex]

Now, Consider Discriminant of the quadratic equation

[tex]\sf \: Discriminant = {b}^{2} - 4ac \\ \\ [/tex]

[tex]\sf \: Discriminant = {( - 3)}^{2} - 4(2)(5) \\ \\ [/tex]

[tex]\sf \: Discriminant = 9 - 40 \\ \\ [/tex]

[tex]\sf \: Discriminant = - 31 \\ \\ [/tex]

[tex]\sf \:  \implies \: Discriminant < 0 \\ \\ [/tex]

[tex]\sf \:  \implies \: Equation \: have \: no \: real \: roots. \\ \\ [/tex]

[tex]\rule{190pt}{2pt}[/tex]

Concept Used :-

Nature of roots :-

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac

Answer:

Given :-

• 2x² - 3x + 5 = 0

To Find :-

• What is the nature of roots of the following quadratic equations.

Solution :-

Given Equation :

[tex]\mapsto \bf 2x^2 - 3x + 5 =\: 0\\[/tex]

By comparing with ax² + bx + c = 0 we get,

• a = 2
• b = - 3
• c = 5

As we know that :

[tex]\bigstar \: \: \sf\boxed{\bold{D =\: b^2 - 4ac}}\: \: \: \bigstar\\[/tex]

where,

• D = Discriminant

According to the question by using the formula we get,

[tex]\implies \sf\boxed{\bold{Discriminant (D) =\: b^2 - 4ac}}\\[/tex]

Given :

• a = 2
• b = - 3
• c = 5

So, by putting those values we get,

[tex]\implies \sf Discriminant (D) =\: (- 3)^2 - 4(2)(5)\\[/tex]

[tex]\implies \sf Discriminant (D) =\: (- 3 \times - 3) - 4(2 \times 5)\\[/tex]

[tex]\implies \sf Discriminant (D) =\: (9) - 4(10)\\[/tex]

[tex]\implies \sf Discriminant (D) =\: (9) - (4 \times 10)\\[/tex]

[tex]\implies \sf Discriminant (D) =\: (9) - (40)\\[/tex]

[tex]\implies \sf Discriminant (D) =\: (9 - 40)\\[/tex]

[tex]\implies \sf Discriminant (D) =\: - 31\\[/tex]

[tex]\implies \sf\bold{\underline{Discriminant (D)\: < 0}}\\[/tex]

[tex]\therefore[/tex] The discriminant of the quadratic equation is - 31 .

Hence, the nature of roots is there will be no real roots .