Let's solve the quadratic equation step-by-step:
Given equation:
[tex]\[ 2x^2 + 1 = 5x \][/tex]
Step 1: Rewrite the equation in standard quadratic form:
[tex]\[ 2x^2 - 5x + 1 = 0 \][/tex]
Step 2: Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the quadratic equation [tex]\(2x^2 - 5x + 1 = 0\)[/tex]:
[tex]\[ a = 2, \, b = -5, \, c = 1 \][/tex]
Step 3: Calculate the discriminant [tex]\(\Delta\)[/tex] of the quadratic equation:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-5)^2 - 4 \cdot 2 \cdot 1 \][/tex]
[tex]\[ \Delta = 25 - 8 \][/tex]
[tex]\[ \Delta = 17 \][/tex]
Step 4: Use the quadratic formula to solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[ x = \frac{-(-5) \pm \sqrt{17}}{2 \cdot 2} \][/tex]
[tex]\[ x = \frac{5 \pm \sqrt{17}}{4} \][/tex]
Therefore, the roots of the equation are:
[tex]\[ x = \frac{5 + \sqrt{17}}{4} \][/tex]
[tex]\[ x = \frac{5 - \sqrt{17}}{4} \][/tex]
By comparing these two solutions with the provided answer choices, we can determine which ones are correct.
Correct Answer:
A. [tex]\( x = \frac{5 - \sqrt{17}}{4} \)[/tex]
B. [tex]\( x = \frac{5 + \sqrt{17}}{4} \)[/tex]
