√-24x-x²=9

To solve the equation \( \sqrt{-24x - x^2} = 9 \), we will first square both sides to eliminate the square root.

1. Square both sides: \[ -24x - x^2 = 81 \]

2. Rearrange the equation: \[ -x^2 - 24x - 81 = 0 \]

3. Multiply the entire equation by -1 to make the leading coefficient positive: \[ x^2 + 24x + 81 = 0 \]

4. Now we can use the quadratic formula to solve for \(x\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 1\), \(b = 24\), and \(c = 81\).

5. First, calculate the discriminant: \[ b^2 - 4ac = 24^2 - 4 \cdot 1 \cdot 81 = 576 - 324 = 252 \]

6. Now apply the quadratic formula: \[ x = \frac{-24 \pm \sqrt{252}}{2} \]

   Simplifying \( \sqrt{252} \): \[ \sqrt{252} = \sqrt{36 \cdot 7} = 6\sqrt{7} \]

7. Substitute back into the formula: \[ x = \frac{-24 \pm 6\sqrt{7}}{2} \]

8. Simplify: \[ x = -12 \pm 3\sqrt{7} \]

Thus, the solutions to the equation \( \sqrt{-24x - x^2} = 9 \) are: \[ x = -12 + 3\sqrt{7} \quad \text{and} \quad x = -12 - 3\sqrt{7} \]

Verification

Finally, it is necessary to check whether these solutions satisfy the original equation, since we squared both sides.

1. For \( x = -12 + 3\sqrt{7} \): \[ -24(-12 + 3\sqrt{7}) - (-12 + 3\sqrt{7})^2 \] Calculate this value and check if it equals 81.

2. For \( x = -12 - 3\sqrt{7} \): Same as above, calculate and check if it matches.

Both solutions must be verified against the original equation since squaring can introduce extraneous solutions.