Roots Nature for Quadratic Equation $3x^{2} – 5x + 2 = 0$ Analysis

To determine the nature of the roots of a given quadratic equation, we analyze its discriminant. The discriminant is a value that tells us whether the roots are real and distinct, real and equal, or complex.

Discriminant Calculation Steps

A standard quadratic equation is represented in the form $ax^{2} + bx + c = 0$, where $a$, $b$, and $c$ are coefficients.

For the given equation, $3x^{2} – 5x + 2 = 0$:

• The coefficient $a$ is 3.
• The coefficient $b$ is -5.
• The coefficient $c$ is 2.

The formula to calculate the discriminant ($\Delta$) is:

$\Delta = b^{2} - 4ac$

Let's substitute the values of $a$, $b$, and $c$ into the formula:

$\Delta = (-5)^{2} - 4(3)(2)$

$\Delta = 25 - 24$

$\Delta = 1$

Interpretation of Discriminant for Root Nature

The value of the discriminant helps us understand the nature of the roots:

• If $\Delta > 0$, the equation has two distinct real roots.
• If $\Delta = 0$, the equation has equal real roots (one real root).
• If $\Delta < 0$, the equation has complex roots (two complex conjugate roots).

In this case, the calculated discriminant is $\Delta = 1$.

Since $1 > 0$, the quadratic equation $3x^{2} – 5x + 2 = 0$ has two distinct real roots.