To solve the quadratic equation [tex]\( x^2 - 5x + 3 = 0 \)[/tex] using the quadratic formula, follow these steps:
Given the general quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = 3 \)[/tex].
1. Calculate the Discriminant:
The discriminant ([tex]\( \Delta \)[/tex]) is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = (-5)^2 - 4(1)(3) \][/tex]
[tex]\[ \Delta = 25 - 12 \][/tex]
[tex]\[ \Delta = 13 \][/tex]
2. Compute the Solutions:
Using the quadratic formula with the calculated discriminant:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute [tex]\( b \)[/tex], the discriminant ([tex]\( \Delta \)[/tex]), and [tex]\( a \)[/tex]:
[tex]\[ x = \frac{-(-5) \pm \sqrt{13}}{2(1)} \][/tex]
[tex]\[ x = \frac{5 \pm \sqrt{13}}{2} \][/tex]
So the solutions to the equation [tex]\( x^2 - 5x + 3 = 0 \)[/tex] are:
[tex]\[
x = \frac{5 + \sqrt{13}}{2} \quad \text{and} \quad x = \frac{5 - \sqrt{13}}{2}
\][/tex]
Therefore, the correct option is:
C. [tex]\( x = \frac{5 \pm \sqrt{13}}{2} \)[/tex]
