Questions: Solve by completing the square and applying the square root property. Express the solution set in exact s c^2 + 16c - 4 = 0 The solution set is .

Solution Steps

To solve the equation \( c^2 + 16c - 4 = 0 \) by completing the square, we first isolate the constant term on one side:

\[ c^2 + 16c = 4 \]

Next, we take half of the coefficient of \( c \) (which is 16), square it, and add it to both sides. Half of 16 is 8, and squaring it gives us 64:

\[ c^2 + 16c + 64 = 4 + 64 \]

This simplifies to:

\[ (c + 8)^2 = 68 \]

Now, we apply the square root property to solve for \( c \):

\[ c + 8 = \pm \sqrt{68} \]

Subtracting 8 from both sides gives us:

\[ c = -8 \pm \sqrt{68} \]

The square root \( \sqrt{68} \) can be simplified:

\[ \sqrt{68} = \sqrt{4 \cdot 17} = 2\sqrt{17} \]

Thus, we can express \( c \) as:

\[ c = -8 \pm 2\sqrt{17} \]

Final Answer