Q1(i):

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (i) $2x^2 – 3x + 5 = 0$

Given: A quadratic equation $2x^2 - 3x + 5 = 0$.

To Find: The nature of the roots of the given quadratic equation and, if real roots exist, to determine their values.

Step 1: Identify the coefficients of the quadratic equation.
A standard quadratic equation is represented as $ax^2 + bx + c = 0$.
Comparing the given equation $2x^2 - 3x + 5 = 0$ with the standard form:
$a = 2$
$b = -3$
$c = 5$

Step 2: Determine the nature of the roots using the Discriminant ($D$).
The discriminant of a quadratic equation is given by the formula:
$D = b^2 - 4ac$
Substituting the values of $a$, $b$, and $c$ into the formula:
$D = (-3)^2 - 4(2)(5)$
$D = 9 - 40$
$D = -31$

Step 3: Analyze the nature of the roots based on the value of $D$.
According to the theory of quadratic equations:
1. If $D > 0$, the equation has two distinct real roots.
2. If $D = 0$, the equation has two equal real roots.
3. If $D < 0$, the equation has no real roots (the roots are complex/imaginary).

Since $D = -31$, and $-31 < 0$, the discriminant is negative.

Step 4: Conclusion regarding the existence of real roots.
Because the discriminant is less than zero ($D < 0$), the square root of the discriminant would result in an imaginary number. Therefore, the quadratic equation $2x^2 - 3x + 5 = 0$ does not possess any real roots.

Final Answer: The discriminant is $-31$, which is less than $0$. Therefore, the given quadratic equation has no real roots.