To solve the equation \( 5 - 2x^2 = -15 \) using inverse operations, we will follow these steps:
Step 1: Isolate the term with \( x^2 \)
First, we need to isolate the term that contains \( x^2 \). We can do this by subtracting 5 from both sides of the equation:
\[ 5 - 2x^2 - 5 = -15 - 5 \]
This simplifies to:
\[ -2x^2 = -20 \]
Step 2: Divide by -2
Next, we divide both sides by -2 to solve for \( x^2 \):
\[ \frac{-2x^2}{-2} = \frac{-20}{-2} \]
This simplifies to:
\[ x^2 = 10 \]
Step 3: Take the square root
Now, we take the square root of both sides to solve for \( x \):
\[ x = \pm\sqrt{10} \]
Step 4: Final answers
The solutions are:
\[ x = \sqrt{10} \quad \text{and} \quad x = -\sqrt{10} \]
Step 5: Check the solutions
Now we will check both solutions in the original equation.
For \( x = \sqrt{10} \):
\[ 5 - 2(\sqrt{10})^2 = 5 - 2(10) = 5 - 20 = -15 \]
This is true.
For \( x = -\sqrt{10} \):
\[ 5 - 2(-\sqrt{10})^2 = 5 - 2(10) = 5 - 20 = -15 \]
This is also true.
Conclusion
Both solutions check out. Therefore, the final answers are:
\[ \boxed{\sqrt{10}} \quad \text{and} \quad \boxed{-\sqrt{10}} \]