If `log_(10)[98+sqrt((x^(3)-x^(2)-12x+36))]=2` and `x` is real then x = …....

Answer

To solve the equation \( \log_{10}\left(98 + \sqrt{x^3 - x^2 - 12x + 36}\right) = 2 \), we will follow these steps:

### Step 1: Convert the logarithmic equation to exponential form.
From the logarithmic equation, we can express it in exponential form:
\[ 98 + \sqrt{x^3 - x^2 - 12x + 36} = 10^2 \]
This simplifies to:
\[ 98 + \sqrt{x^3 - x^2 - 12x + 36} = 100 \]

### Step 2: Isolate the square root term.
Next, we isolate the square root:
\[ \sqrt{x^3 - x^2 - 12x + 36} = 100 - 98 \]
This simplifies to:
\[ \sqrt{x^3 - x^2 - 12x + 36} = 2 \]

### Step 3: Square both sides to eliminate the square root.
Now we square both sides:
\[ x^3 - x^2 - 12x + 36 = 2^2 \]
This simplifies to:
\[ x^3 - x^2 - 12x + 36 = 4 \]

### Step 4: Rearrange the equation.
Rearranging gives us:
\[ x^3 - x^2 - 12x + 36 - 4 = 0 \]
This simplifies to:
\[ x^3 - x^2 - 12x + 32 = 0 \]

### Step 5: Factor the cubic polynomial.
We can try to find rational roots using the Rational Root Theorem. Testing \( x = -4 \):
\[ (-4)^3 - (-4)^2 - 12(-4) + 32 = -64 - 16 + 48 + 32 = 0 \]
Since \( x = -4 \) is a root, we can factor the polynomial as \( (x + 4)(\text{quadratic}) \).

### Step 6: Perform polynomial long division.
Dividing \( x^3 - x^2 - 12x + 32 \) by \( x + 4 \):
1. Divide the leading term: \( x^3 \div x = x^2 \)
2. Multiply: \( x^2(x + 4) = x^3 + 4x^2 \)
3. Subtract: \( (-x^2 - 4x^2) = -5x^2 \)
4. Bring down: \( -5x^2 - 12x + 32 \)
5. Repeat: \( -5x^2 \div x = -5x \)
6. Multiply: \( -5x(x + 4) = -5x^2 - 20x \)
7. Subtract: \( (-12x + 20x) = 8x + 32 \)
8. Repeat: \( 8x \div x = 8 \)
9. Multiply: \( 8(x + 4) = 8x + 32 \)
10. Subtract: \( 0 \)

Thus, we have:
\[ x^3 - x^2 - 12x + 32 = (x + 4)(x^2 - 5x + 8) \]

### Step 7: Solve the quadratic equation.
Now we solve the quadratic equation:
\[ x^2 - 5x + 8 = 0 \]
Using the discriminant:
\[ D = b^2 - 4ac = (-5)^2 - 4(1)(8) = 25 - 32 = -7 \]
Since the discriminant is negative, there are no real solutions from this quadratic.

### Step 8: Conclusion.
The only real solution comes from \( x + 4 = 0 \):
\[ x = -4 \]

Thus, the final answer is:
\[ \boxed{-4} \]