To solve the quadratic equation [tex]\( 2x^2 - 5x + 1 = 3 \)[/tex], follow these detailed steps:
1. Simplify the Equation:
- Start by moving the constant term on the right side of the equation to the left side.
[tex]\[
2x^2 - 5x + 1 - 3 = 0
\][/tex]
- Simplify the equation:
[tex]\[
2x^2 - 5x - 2 = 0
\][/tex]
2. Identify Coefficients:
- The quadratic equation is in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = -5 \)[/tex]
- [tex]\( c = -2 \)[/tex]
3. Apply the Quadratic Formula:
- The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
4. Substitute the Coefficients into the Formula:
- Substitute [tex]\( a = 2 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = -2 \)[/tex] into the formula:
[tex]\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2}
\][/tex]
5. Simplify the Expression:
- Simplify the terms inside the formula:
[tex]\[
x = \frac{5 \pm \sqrt{25 + 16}}{4}
\][/tex]
- Further simplify under the square root:
[tex]\[
x = \frac{5 \pm \sqrt{41}}{4}
\][/tex]
6. Express the Solutions:
- Therefore, the solutions are:
[tex]\[
x = \frac{5 + \sqrt{41}}{4} \quad \text{and} \quad x = \frac{5 - \sqrt{41}}{4}
\][/tex]
Thus, the correct solution among the given choices is:
[tex]\[
x = \frac{5}{4} \pm \frac{\sqrt{41}}{4}
\][/tex]
