To solve the quadratic equation [tex]\( \frac{-6 \pm \sqrt{6^2-4(3 \cdot 9)}}{2 \cdot 3} \)[/tex], let’s break it down step-by-step:
1. Identify the coefficients:
In the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the given coefficients are:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = -6 \)[/tex]
- [tex]\( c = 9 \)[/tex]
2. Calculate the discriminant [tex]\( \Delta \)[/tex]:
The discriminant is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
Plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = (-6)^2 - 4 \cdot 3 \cdot 9 \][/tex]
[tex]\[ \Delta = 36 - 108 \][/tex]
[tex]\[ \Delta = -72 \][/tex]
3. Substitute the discriminant back into the quadratic formula:
The quadratic formula is [tex]\( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex].
Now plug in the values of [tex]\( b \)[/tex], [tex]\(\Delta\)[/tex], and [tex]\( a \)[/tex]:
[tex]\[ x = \frac{-(-6) \pm \sqrt{-72}}{2 \cdot 3} \][/tex]
[tex]\[ x = \frac{6 \pm \sqrt{-72}}{6} \][/tex]
4. Simplify the expression under the square root:
Since the discriminant is negative, we will have complex roots. Rewrite [tex]\(\sqrt{-72}\)[/tex] as [tex]\( \sqrt{72}i\)[/tex], where [tex]\( i \)[/tex] is the imaginary unit ([tex]\( i = \sqrt{-1} \)[/tex]).
[tex]\[ \sqrt{72} \][/tex]
[tex]\[ \sqrt{72} = \sqrt{36 \cdot 2} \][/tex]
[tex]\[ \sqrt{72} = 6\sqrt{2} \][/tex]
Thus,
[tex]\[ \sqrt{-72} = 6\sqrt{2}i \][/tex]
5. Divide by [tex]\( 2a \)[/tex] and simplify:
[tex]\[ x = \frac{6 \pm 6\sqrt{2}i}{6} \][/tex]
[tex]\[ x = 1 \pm \sqrt{2}i \][/tex]
Therefore, the two solutions to the quadratic equation [tex]\( 3x^2 - 6x + 9 = 0 \)[/tex] are:
[tex]\[ x_1 = 1 + \sqrt{2}i \][/tex]
[tex]\[ x_2 = 1 - \sqrt{2}i \][/tex]
These roots are complex because the discriminant was negative, and they come in a conjugate pair.
