To solve the quadratic equation [tex]\( x^2 - 12x + 59 = 0 \)[/tex], we should use the quadratic formula. The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For our equation [tex]\( x^2 - 12x + 59 = 0 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -12 \)[/tex]
- [tex]\( c = 59 \)[/tex]
First, we calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-12)^2 - 4 \cdot 1 \cdot 59 \][/tex]
[tex]\[ \Delta = 144 - 236 \][/tex]
[tex]\[ \Delta = -92 \][/tex]
Since the discriminant is negative ([tex]\(\Delta = -92\)[/tex]), we know that the roots will be complex numbers.
Now we can find the roots using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex]:
[tex]\[ x = \frac{12 \pm \sqrt{-92}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{12 \pm \sqrt{92}i}{2} \][/tex]
We can simplify [tex]\( \sqrt{92} \)[/tex]:
[tex]\[ \sqrt{92} = \sqrt{4 \cdot 23} = 2\sqrt{23} \][/tex]
So the equation for [tex]\( x \)[/tex] becomes:
[tex]\[ x = \frac{12 \pm 2\sqrt{23}i}{2} \][/tex]
[tex]\[ x = 6 \pm \sqrt{23}i \][/tex]
Thus, the roots of the quadratic equation [tex]\( x^2 - 12x + 59 = 0 \)[/tex] are:
[tex]\[ x = 6 \pm \sqrt{23}i \][/tex]
Therefore, the correct answer is:
[tex]\[ x = 6 \pm \sqrt{23}i \][/tex]
