To solve the quadratic equation [tex]\( x^2 - 5x + 2 = 0 \)[/tex], we will use the quadratic formula, which is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In this equation, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]. For the given quadratic equation [tex]\(x^2 - 5x + 2 = 0\)[/tex], the coefficients are:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -5\)[/tex]
- [tex]\(c = 2\)[/tex]
First, we need to calculate the discriminant, which is the part under the square root in the quadratic formula:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \text{Discriminant} = (-5)^2 - 4 \cdot 1 \cdot 2 \][/tex]
[tex]\[ \text{Discriminant} = 25 - 8 \][/tex]
[tex]\[ \text{Discriminant} = 17 \][/tex]
Since the discriminant is positive ([tex]\(17\)[/tex]), we will have two real roots. Now we substitute the coefficients and the discriminant into the quadratic formula to find the roots:
[tex]\[ x = \frac{-b \pm \sqrt{17}}{2a} \][/tex]
Given [tex]\(a = 1\)[/tex] and [tex]\(b = -5\)[/tex]:
[tex]\[ x = \frac{-(-5) \pm \sqrt{17}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{5 \pm \sqrt{17}}{2} \][/tex]
Now, we calculate the two roots separately:
1. For the plus sign ([tex]\(+\)[/tex]):
[tex]\[ x_1 = \frac{5 + \sqrt{17}}{2} \][/tex]
The numerical solution is approximately:
[tex]\[ x_1 \approx 4.561552812808831 \][/tex]
2. For the minus sign ([tex]\(-\)[/tex]):
[tex]\[ x_2 = \frac{5 - \sqrt{17}}{2} \][/tex]
The numerical solution is approximately:
[tex]\[ x_2 \approx 0.4384471871911697 \][/tex]
Thus, the roots of the quadratic equation [tex]\( x^2 - 5x + 2 = 0 \)[/tex] are approximately:
[tex]\[ x_1 \approx 4.561552812808831 \][/tex]
[tex]\[ x_2 \approx 0.4384471871911697 \][/tex]
These are the two solutions for the quadratic equation.
