To solve the equation \( 4 - x^2 = -16 \), we will follow the steps using algebraic operations.
Step 1: Isolate the term with \( x^2 \)
Start by moving 4 to the right side of the equation.
\[ 4 - x^2 = -16 \]
Subtracting 4 from both sides:
\[ -x^2 = -16 - 4 \]
This simplifies to:
\[ -x^2 = -20 \]
Step 2: Multiply both sides by -1
To eliminate the negative sign in front of \( x^2 \), multiply both sides of the equation by -1:
\[ x^2 = 20 \]
Step 3: Solve for \( x \) by taking the square root
To solve for \( x \), take the square root of both sides. Remember, taking the square root of a number gives both a positive and negative solution.
\[ x = \pm \sqrt{20} \]
Step 4: Simplify \( \sqrt{20} \)
Now, simplify \( \sqrt{20} \):
\[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \]
Step 5: Write the final solutions
Now, incorporate the positive and negative solutions:
\[ x = \pm 2\sqrt{5} \]
Thus, the final answer is:
\[ x = 2\sqrt{5} \quad \text{or} \quad x = -2\sqrt{5} \]
These are the solutions to the equation \( 4 - x^2 = -16 \).
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