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The quadratic equation given is:

\[ 2x^2 - 5x + 1 = 0 \]

To find the sum, product, and roots of the equation, we can use the following methods:

### 1. **Sum of the Roots:**
For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of the roots is given by:

\[
\text{Sum} = -\frac{b}{a}
\]

Here, \( a = 2 \), \( b = -5 \), so:

\[
\text{Sum} = -\left(-\frac{5}{2}\right) = \frac{5}{2}
\]

### 2. **Product of the Roots:**
The product of the roots is given by:

\[
\text{Product} = \frac{c}{a}
\]

Here, \( c = 1 \), so:

\[
\text{Product} = \frac{1}{2}
\]

### 3. **Roots of the Equation:**
The roots can be found using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substituting the values:

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(1)}}{2(2)}
\]

\[
x = \frac{5 \pm \sqrt{25 - 8}}{4}
\]

\[
x = \frac{5 \pm \sqrt{17}}{4}
\]

So, the roots are:

\[
x_1 = \frac{5 + \sqrt{17}}{4}, \quad x_2 = \frac{5 - \sqrt{17}}{4}
\]

### Summary:
- **Sum of the roots:** \( \frac{5}{2} \)
- **Product of the roots:** \( \frac{1}{2} \)
- **Roots:** \( \frac{5 + \sqrt{17}}{4} \) and \( \frac{5 - \sqrt{17}}{4} \)

Would you like any more details or have any questions?

Here are 5 related questions you might find helpful:
1. How can the discriminant be used to determine the nature of the roots?
2. What is the significance of the sum and product of the roots in a quadratic equation?
3. How can you factorize the quadratic equation \( 2x^2 - 5x + 1 = 0 \) based on its roots?
4. How would the sum and product change if the quadratic equation were \( 2x^2 - 5x - 1 = 0 \)?
5. Can you derive the quadratic formula from completing the square?

**Tip:** Remember, the discriminant \( b^2 - 4ac \) is key to determining whether the roots are real and distinct, real and repeated, or complex.

2x^2 - 5x + 1 = 0. Find the sum, product, and roots of the equation.

Learn how to solve the quadratic equation 2x^2 - 5x + 1 = 0 by finding the sum, product, and roots. This problem covers essential algebraic concepts, including the quadratic formula and the calculation of sum and product of roots. Suitable for students in Grades 7-9.

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