Solve for x in the equation 2 x squared minus 5 x + 1 = 3.
x = five-halves plus-or-minus StartFraction StartRoot 29 EndRoot Over 2 EndFraction
x = five-halves plus-or-minus StartFraction StartRoot 41 EndRoot Over 4 EndFraction
x = five-fourths plus-or-minus StartFraction StartRoot 29 EndRoot Over 2 EndFraction
x = five-fourths plus-or-minus StartFraction StartRoot 41 EndRoot Over 4 EndFraction

To solve for \(x\) in the equation \( 2x^2 - 5x + 1 = 3 \), we first bring all terms to one side of the equation:

\[
2x^2 - 5x + 1 - 3 = 0 \quad \Rightarrow \quad 2x^2 - 5x - 2 = 0.
\]

This is a quadratic equation of the form \( ax^2 + bx + c = 0 \), where:
- \(a = 2\),
- \(b = -5\),
- \(c = -2\).

We can now solve this using the quadratic formula:

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

Substituting in the values for \(a\), \(b\), and \(c\):

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-2)}}{2(2)}
\]
\[
x = \frac{5 \pm \sqrt{25 + 16}}{4}
\]
\[
x = \frac{5 \pm \sqrt{41}}{4}.
\]

Thus, the solutions are:

\[
x = \frac{5 \pm \sqrt{41}}{4}.
\]

So the correct solution is:

\[
x = \frac{5}{4} \pm \frac{\sqrt{41}}{4}.
\]

This matches the third option:

\[
x = \frac{5}{4} \pm \frac{\sqrt{41}}{4}.
\]

### Answer:
\( x = \frac{5}{4} \pm \frac{\sqrt{41}}{4} \).

Would you like more details on how the quadratic formula works or any other step? 

Here are five related questions for further exploration:
1. How do you derive the quadratic formula?
2. What are the conditions for real and complex solutions in quadratic equations?
3. How would the solution change if the discriminant were negative?
4. Can you solve a quadratic equation by completing the square?
5. What are some real-world applications of quadratic equations?

**Tip:** Always check the discriminant \(\Delta = b^2 - 4ac\) to quickly determine the nature of the roots (real, repeated, or complex).

Solve for x in the equation 2x^2 - 5x + 1 = 3. x = five-halves plus-or-minus StartFraction StartRoot 29 EndRoot Over 2 EndFraction x = five-halves plus-or-minus StartFraction StartRoot 41 EndRoot Over 4 EndFraction x = five-fourths plus-or-minus StartFraction StartRoot 29 EndRoot Over 2 EndFraction x = five-fourths plus-or-minus StartFraction StartRoot 41 EndRoot Over 4 EndFraction

Learn how to solve the quadratic equation 2x^2 - 5x + 1 = 3 step by step using the quadratic formula. This algebra problem explains the process of solving for x, using key formulas like the quadratic formula and discriminant. Perfect for students in Grades 9-10.

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