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Unlock the power of precision and clarity in problem-solving with our expertly crafted prompt template. Designed for the intellectually curious and the academically driven, this template is your gateway to exploring the intricacies of mathematics and the wonders of scientific phenomena. Whether you're delving into complex equations or unraveling the threads of groundbreaking concepts, our prompt offers a structured approach that guides you to enlightenment. Ideal for students, educators.

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Sure! Let's go through the step-by-step solution for the given quadratic equation \( 3x^2 - 5x + 2 = 0 \).
### Step 1: Identify the coefficients
For the quadratic equation \( ax^2 + bx + c = 0 \), identify the coefficients \( a \), \( b \), and \( c \):
- \( a = 3 \)
- \( b = -5 \)
- \( c = 2 \)
### Step 2: Calculate the discriminant
The discriminant (\( \Delta \)) of a quadratic equation is given by:
\[ \Delta = b^2 - 4ac \]
Substitute the coefficients:
\[ \Delta = (-5)^2 - 4(3)(2) \]
\[ \Delta = 25 - 24 \]
\[ \Delta = 1 \]
### Step 3: Determine the nature of the roots
The discriminant helps determine the nature of the roots of the quadratic equation:
- If \( \Delta > 0 \): The equation has two distinct real roots.
- If \( \Delta = 0 \): The equation has exactly one real root (a repeated root).
- If \( \Delta < 0 \): The equation has two complex conjugate roots.
Since \( \Delta = 1 \) is positive, the equation has two distinct real roots.
### Step 4: Calculate the roots
The roots of the quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
Substitute the coefficients and the value of the discriminant:
\[ x = \frac{-(-5) \pm \sqrt{1}}{2(3)} \]
\[ x = \frac{5 \pm 1}{6} \]
This gives us two roots:
\[ x_1 = \frac{5 + 1}{6} = \frac{6}{6} = 1 \]
\[ x_2 = \frac{5 - 1}{6} = \frac{4}{6} = \frac{2}{3} \]
### Summary
The roots of the quadratic equation \( 3x^2 - 5x + 2 = 0 \) are \( x = 1 \) and \( x = \frac{2}{3} \).
### Significance of the Discriminant
- **Positive Discriminant (\( \Delta > 0 \))**: Indicates two distinct real roots.
- **Zero Discriminant (\( \Delta = 0 \))**: Indicates exactly one real root (a repeated root).
- **Negative Discriminant (\( \Delta < 0 \))**: Indicates two complex conjugate roots.
Let's briefly illustrate the other cases for completeness:
#### Zero Discriminant Example
Consider the equation \( x^2 - 2x + 1 = 0 \):
- \( a = 1 \), \( b = -2 \), \( c = 1 \)
- \( \Delta = (-2)^2 - 4(1)(1) = 4 - 4 = 0 \)
With \( \Delta = 0 \):
\[ x = \frac{-(-2) \pm \sqrt{0}}{2(1)} = \frac{2 \pm 0}{2} = 1 \]
This equation has one real root: \( x = 1 \).
#### Negative Discriminant Example
Consider the equation \( x^2 + x + 1 = 0 \):
- \( a = 1 \), \( b = 1 \), \( c = 1 \)
- \( \Delta = 1^2 - 4(1)(1) = 1 - 4 = -3 \)
With \( \Delta = -3 \):
\[ x = \frac{-1 \pm \sqrt{-3}}{2(1)} = \frac{-1 \pm \sqrt{3}i}{2} \]
This equation has two complex roots:
\[ x_1 = \frac{-1 + \sqrt{3}i}{2} \]
\[ x_2 = \frac{-1 - \sqrt{3}i}{2} \]
These steps illustrate how to determine and find the roots of a quadratic equation based on the discriminant.

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