To find the simplified form of the solutions to the quadratic equation [tex]\(x^2 - 10x + 2 = 0\)[/tex], we start with the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \][/tex]
where [tex]\(a = 1\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = 2\)[/tex].
Plugging these values into the quadratic formula, we get:
[tex]\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}. \][/tex]
Simplifying this, we have:
[tex]\[ x = \frac{10 \pm \sqrt{100 - 8}}{2}. \][/tex]
Next, we simplify what's inside the square root:
[tex]\[ x = \frac{10 \pm \sqrt{92}}{2}. \][/tex]
So, the solutions to the equation are:
[tex]\[ x = \frac{10 + \sqrt{92}}{2} \quad \text{and} \quad x = \frac{10 - \sqrt{92}}{2}. \][/tex]
To further simplify these, note that:
[tex]\[ \sqrt{92} \approx 9.591663046625438. \][/tex]
Now, we calculate each solution individually:
1. For [tex]\(x_1\)[/tex]:
[tex]\[ x_1 = \frac{10 + 9.591663046625438}{2} \approx \frac{19.591663046625438}{2} \approx 9.79583152331272. \][/tex]
2. For [tex]\(x_2\)[/tex]:
[tex]\[ x_2 = \frac{10 - 9.591663046625438}{2} \approx \frac{0.408336953374562}{2} \approx 0.2041684766872809. \][/tex]
Thus, the simplified forms of the solutions are:
[tex]\[ x_1 \approx 9.79583152331272 \][/tex]
[tex]\[ x_2 \approx 0.2041684766872809. \][/tex]
