Answer

Verified

454.2k+ views

**Hint:**The nature of the roots depends on the value of the discriminant of the quadratic equation.

$a{x^2} + bx + c = 0$, where $a \ne 0$

Find the Discriminant, $D = {b^2} - 4ac$ , of the given quadratic equation, and check the sign (i.e. positive or negative or zero) to know if there are two solutions or one solution or no solution.

**Complete step-by-step answer:**

Step 1: Given the quadratic equation:

$2{x^2} - 3x + 5 = 0$

On comparing with standard quadratic equation: $a{x^2} + bx + c = 0$, where $a \ne 0$

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

Step 2: Find discriminant:

$D = {b^2} - 4ac$

$D = {\left( { - 3} \right)^2} - 4 \times 2 \times 5$

$

\Rightarrow {\text{ }} = 9 - 40 \\

\Rightarrow {\text{ }} = - 31 \\

$

Step 3: Check the sign of discriminant:

$D < 0$

Hence, the roots are imaginary.

Final answer: The roots of $2{x^2} - 3x + 5 = 0$ are not real. Thus the correct option is (C).

**Additional Information:**

Roots of the quadratic equation is given by:

Quadratic equation: $a{x^2} + bx + c = 0$, where $a \ne 0$

Roots: \[x = \dfrac{{ - b \pm \sqrt D }}{{2a}}\]

The imaginary roots of the given quadratic equation are:

$2{x^2} - 3x + 5 = 0$

D = -31

\[x = \dfrac{{ - b \pm \sqrt D }}{{2a}}\]

$

{\text{ }}x = \dfrac{{ - \left( { - 3} \right) \pm \sqrt {\left( { - 31} \right)} }}{{2\left( 2 \right)}} \\

\Rightarrow {\text{ }} = \dfrac{{3 \pm {\text{i}}\sqrt {31} }}{4} \\

$

**Note:**For quadratic equation: $a{x^2} + bx + c = 0$, where $a \ne 0$

Let $y = f\left( x \right) = a{x^2} + bx + c = 0$

Discriminant, $D = {b^2} - 4ac$

A discriminant of zero indicates that the quadratic has a repeated real number solution.

i.e. $D = 0$ , roots are real and equal.

$ \Rightarrow {b^2} - 4ac = 0$

A positive discriminant indicates that the quadratic has two distinct real number solutions.

i.e. $D > 0$ , roots are real and unequal.

$ \Rightarrow {b^2} - 4ac > 0$

A negative discriminant indicates that neither of the solutions is real numbers.

And if D < 0, as in the case of the given question, roots are imaginary.

$ \Rightarrow {b^2} - 4ac < 0$

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

How do you graph the function fx 4x class 9 maths CBSE

Which are the Top 10 Largest Countries of the World?

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

1 ton equals to A 100 kg B 1000 kg C 10 kg D 10000 class 11 physics CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

The largest tea producing country in the world is A class 10 social science CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE