Question

Check whether \[2{x^2} - 3x + 5 = 0\] has real roots or not.
A.The equation has real roots.
B.The equation has no real roots.
C.Data insufficient.
D.None of these.

Answer 1

Hint: The roots of any equation are the values of the variable for which it satisfies the equation. The total number of roots of any one variable equation is equal to the degree of the equation, and degree of equation is the highest power of the variable. Here in the given question the degree of the equation is \[2\] and so there are two possible roots of this equation.

Complete step-by-step answer:
Given equation is
 \[2{x^2} - 3x + 5 = 0\]
This is the quadratic equation. We have to find the nature of the roots of this equation. The nature of the roots of the quadratic equation is obtained by the discriminant of the root.
The discriminant of the quadratic equation of the form \[a{x^2} + bx + c = 0\] is given by
 \[D = {b^2} - 4ac\]
Where,
 \[a = {\text{ }}Coefficient{\text{ }}of{\text{ }}{x^2}\]
 \[b = {\text{ }}coefficient{\text{ }}of{\text{ }}x\]
 \[c = {\text{ }}constant{\text{ }}term\]
The nature of the two roots of the quadratic equation are:
Real and equal iff Discriminant, \[D = 0\]
Real and distinct iff discriminant, \[D > 0\]
And, No real roots or have complex roots if discriminant, \[D < 0\]
Here given quadratic equation is \[2{x^2} - 3x + 5 = 0\]
So, we have
 \[\begin{array}{*{20}{l}}
  {a = 2} \\
  {b = - 3} \\
  {c = 5}
\end{array}\]
And so discriminant, \[D = {b^2} - 4ac\]
Substituting the values
 \[D = {\left( { - 3} \right)^2} - 4 \times 2 \times 5\]
On squaring and simplifying
 \[D = 9 - 40\]
 \[D = - 36 < 0\]
Clearly Discriminant, \[D < 0\]
Hence the given quadratic equation has no real roots
Hence Option B is correct.
So, the correct answer is “Option B”.

Note: The roots of the equation are either real or complex. Complex roots are always in even numbers as complex numbers always exist with its conjugate. Discriminant of the quadratic equation in the simplest method to know the nature of the roots. A quadratic equation could have either real roots or complex roots. It cannot have both types of roots but equations of higher order could have real as well as complex roots at the same time.