To solve the quadratic equation \( x^2 + 12x - 14 = 0 \), we can use the quadratic formula:
\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]
In this equation, \( a = 1 \), \( b = 12 \), and \( c = -14 \).
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 12^2 - 4 \cdot 1 \cdot (-14) = 144 + 56 = 200 \]
Next, we substitute the values into the quadratic formula:
\[ x = \frac{{-12 \pm \sqrt{200}}}{2 \cdot 1} \]
Since \( \sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2} \), we can simplify:
\[ x = \frac{{-12 \pm 10\sqrt{2}}}{2} \]
Now, we simplify further:
\[ x = \frac{{-12}}{2} \pm \frac{{10\sqrt{2}}}{2} \] \[ x = -6 \pm 5\sqrt{2} \]
Thus, the two solutions are:
\[ x = -6 + 5\sqrt{2} \quad \text{and} \quad x = -6 - 5\sqrt{2} \]
These are the solutions to the equation \( x^2 + 12x - 14 = 0 \).