\( x = \frac { - c \pm \sqrt { 6 ^ { 2 } - 4 ( 1 ) ( 6 ) } } { 2 ( 1 ) } \)

Let's simplify the given quadratic formula step by step: \[ x = \frac{ -c \pm \sqrt{6^2 - 4 \cdot 1 \cdot 6 } }{2 \cdot 1} \] ### Step 1: Calculate the Discriminant The discriminant (\( \Delta \)) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by: \[ \Delta = b^2 - 4ac \] In your expression: - \( a = 1 \) - \( b = c \) - \( c = 6 \) Plugging these values into the discriminant formula: \[ \Delta = c^2 - 4 \cdot 1 \cdot 6 = c^2 - 24 \] ### Step 2: Substitute the Discriminant Back into the Formula \[ x = \frac{ -c \pm \sqrt{c^2 - 24} }{2} \] ### Step 3: Simplify the Expression The expression inside the square root, \( c^2 - 24 \), doesn't simplify further unless you have specific values for \( c \). Therefore, the simplified form of the solution is: \[ x = \frac{ -c \pm \sqrt{c^2 - 24} }{2} \] ### Final Answer \[ x = \frac{ -c \pm \sqrt{c^2 - 24} }{2} \] This represents the two possible solutions for \( x \) in terms of \( c \) for the quadratic equation you've provided.