To solve the quadratic equation [tex]\(4x^2 - x + 9 = 0\)[/tex], we start with identifying the coefficients of the quadratic equation in standard form [tex]\(ax^2 + bx + c = 0\)[/tex]:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = -1\)[/tex]
- [tex]\(c = 9\)[/tex]
Next, we calculate the discriminant [tex]\(\Delta\)[/tex] using the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values, we get:
[tex]\[ \Delta = (-1)^2 - 4 \cdot 4 \cdot 9 \][/tex]
[tex]\[ \Delta = 1 - 144 \][/tex]
[tex]\[ \Delta = -143 \][/tex]
Since the discriminant is less than zero, the quadratic equation has complex roots. We use the quadratic formula to find these roots:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting the given values and discriminant:
[tex]\[ x = \frac{-(-1) \pm \sqrt{-143}}{2 \cdot 4} \][/tex]
[tex]\[ x = \frac{1 \pm \sqrt{-143}}{8} \][/tex]
Since [tex]\(\sqrt{-143} = i\sqrt{143}\)[/tex] (where [tex]\(i\)[/tex] is the imaginary unit), we substitute and get:
[tex]\[ x = \frac{1 \pm i\sqrt{143}}{8} \][/tex]
Thus, the solutions to the quadratic equation are:
[tex]\[ x = \frac{1 + i\sqrt{143}}{8} \][/tex]
[tex]\[ x = \frac{1 - i\sqrt{143}}{8} \][/tex]
Comparing with the options provided:
- Option A: [tex]\(x = \frac{1 + i\sqrt{143}}{8}\)[/tex] or [tex]\(x = \frac{1 - i\sqrt{143}}{8}\)[/tex]
So, the correct answer is:
A. [tex]\( x = \frac{1 + i\sqrt{143}}{8} \)[/tex] or [tex]\( x = \frac{1 - i\sqrt{143}}{8} \)[/tex]
