What is the Nature of the Roots of the Quadratic Equation 9x² + 25 = 30x?

The nature of the roots of the quadratic equation 9x² + 25 = 30x is real and equal.

Understanding the Nature of Roots

The nature of the roots of a quadratic equation ax² + bx + c = 0 (where a ≠ 0) is determined by a specific value called the discriminant. This discriminant helps us predict whether the roots will be real, distinct, equal, or complex without actually solving the equation.

The Discriminant (Δ)

The discriminant is calculated using the formula:

Δ = b² - 4ac

Where:

a is the coefficient of the x² term.
b is the coefficient of the x term.
c is the constant term.

The value of the discriminant dictates the nature of the roots as follows:

Discriminant Value	Nature of Roots
Δ > 0	Real and Distinct (Unequal) Roots. There are two unique real number solutions.
Δ = 0	Real and Equal Roots. There is exactly one real number solution (a repeated root).
Δ < 0	No Real Roots (Complex/Imaginary) Roots. There are two complex conjugate solutions.

Analyzing the Equation: 9x² + 25 = 30x

To determine the nature of the roots for the given quadratic equation, 9x² + 25 = 30x, we must first convert it to the standard form ax² + bx + c = 0 and then calculate its discriminant.

Step-by-Step Calculation

Let's break down the process:

Step 1: Convert to Standard Form
The given equation is 9x² + 25 = 30x.
To write it in the standard form ax² + bx + c = 0, we move all terms to one side:
9x² - 30x + 25 = 0
Step 2: Identify Coefficients
From the standard form 9x² - 30x + 25 = 0, we can identify the coefficients:
- a = 9
- b = -30
- c = 25
Step 3: Calculate the Discriminant
Now, substitute these values into the discriminant formula Δ = b² - 4ac:
Δ = (-30)² - 4(9)(25)
Δ = 900 - 4(225)
Δ = 900 - 900
Δ = 0
Step 4: Determine Nature of Roots
Since the discriminant Δ = 0, according to the table above, the roots of the quadratic equation 9x² + 25 = 30x are real and equal.

Practical Insight: Perfect Square Trinomial

It's also interesting to note that the expression 9x² - 30x + 25 is a perfect square trinomial. It can be factored as (3x - 5)².

If (3x - 5)² = 0, then:
3x - 5 = 0
3x = 5
x = 5/3

This confirms that there is exactly one real root, which is repeated, thus aligning with the "real and equal" nature derived from the discriminant.