To find the solutions of the equation [tex]\( x^2 + 6x - 6 = 10 \)[/tex], we begin by rewriting it in standard form.
First, we subtract 10 from both sides:
[tex]\[ x^2 + 6x - 6 - 10 = 0 \][/tex]
This simplifies to:
[tex]\[ x^2 + 6x - 16 = 0 \][/tex]
Now we have a quadratic equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 6 \)[/tex], and [tex]\( c = -16 \)[/tex].
To solve this quadratic equation, we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Plugging in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-16)}}{2(1)} \][/tex]
Simplify inside the square root:
[tex]\[ x = \frac{-6 \pm \sqrt{36 + 64}}{2} \][/tex]
[tex]\[ x = \frac{-6 \pm \sqrt{100}}{2} \][/tex]
Taking the square root of 100:
[tex]\[ x = \frac{-6 \pm 10}{2} \][/tex]
This gives us two possible solutions:
[tex]\[ x = \frac{-6 + 10}{2} = \frac{4}{2} = 2 \][/tex]
and
[tex]\[ x = \frac{-6 - 10}{2} = \frac{-16}{2} = -8 \][/tex]
Therefore, the solutions to the equation [tex]\( x^2 + 6x - 6 = 10 \)[/tex] are:
[tex]\[ x = -8 \text{ or } x = 2 \][/tex]
So the correct choice is:
[tex]\[ x = -8 \text{ or } x = 2 \][/tex]
