To solve the quadratic equation [tex]\(x^2 + 20 = 2x\)[/tex], follow these steps:
1. Rearrange the equation into standard form [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[x^2 + 20 = 2x\][/tex]
[tex]\[x^2 - 2x + 20 = 0\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = 20\)[/tex].
2. Use the quadratic formula to solve for [tex]\(x\)[/tex]:
The quadratic formula is given by:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
3. Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
[tex]\[b = -2, \, a = 1, \, c = 20\][/tex]
[tex]\[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 20}}{2 \cdot 1}\][/tex]
[tex]\[x = \frac{2 \pm \sqrt{4 - 80}}{2}\][/tex]
[tex]\[x = \frac{2 \pm \sqrt{-76}}{2}\][/tex]
4. Simplify the square root of the discriminant:
Since the discriminant is negative, the solutions will be complex.
[tex]\[\sqrt{-76} = \sqrt{-(4 \cdot 19)} = \sqrt{-1 \cdot 4 \cdot 19} = 2i\sqrt{19}\][/tex]
5. Substitute the simplified square root back into the formula:
[tex]\[x = \frac{2 \pm 2i \sqrt{19}}{2}\][/tex]
[tex]\[x = 1 \pm i \sqrt{19}\][/tex]
So the solutions for [tex]\(x\)[/tex] are:
[tex]\[x = 1 \pm i \sqrt{19}\][/tex]
The correct answer is:
[tex]\[ \boxed{1 \pm i \sqrt{19}} \][/tex]
