How many real solutions of the equation `6x^(2)-77[x]+147=0`, where `[x]` is the integral part of `x`?

Explanation

To solve the equation \( 6x^2 - 77[x] + 147 = 0 \), where \([x]\) is the integral part of \(x\), we will proceed step by step.

### Step 1: Understand the equation
The equation can be rewritten as:
\[ 6x^2 + 147 = 77[x] \]
This means that the left-hand side must equal an integer since \([x]\) is an integer.

### Step 2: Set up the quadratic equation
We can express the equation as:
\[ 6x^2 - 77[x] + 147 = 0 \]
This is a quadratic equation in terms of \(x\).

### Step 3: Analyze the quadratic equation
For the quadratic equation \(6x^2 - 77[x] + 147 = 0\) to have real solutions, the discriminant must be non-negative. The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:
\[ D = b^2 - 4ac \]
In our case, \(a = 6\), \(b = -77[x]\), and \(c = 147\). Thus, the discriminant becomes:
\[ D = (-77[x])^2 - 4 \cdot 6 \cdot 147 \]

### Step 4: Calculate the discriminant
Calculating the discriminant:
\[ D = 5929[x]^2 - 3528 \]
For \(D\) to be non-negative:
\[ 5929[x]^2 - 3528 \geq 0 \]
This simplifies to:
\[ 5929[x]^2 \geq 3528 \]
\[ [x]^2 \geq \frac{3528}{5929} \]
Calculating the right-hand side:
\[ [x]^2 \geq 0.595 \]
Taking the square root:
\[ |[x]| \geq \sqrt{0.595} \approx 0.772 \]

### Step 5: Determine possible integer values
Since \([x]\) is an integer, the smallest integer satisfying this condition is \([x] = 1\) or \([x] = -1\).

### Step 6: Check integer values of \([x]\)
We will check the integer values \([x] = 1\) and \([x] = -1\):

1. **For \([x] = 1\)**:
\[ 6x^2 - 77(1) + 147 = 0 \Rightarrow 6x^2 + 70 = 0 \]
This has no real solutions since the left side is always positive.

2. **For \([x] = -1\)**:
\[ 6x^2 - 77(-1) + 147 = 0 \Rightarrow 6x^2 + 77 + 147 = 0 \Rightarrow 6x^2 + 224 = 0 \]
This also has no real solutions since the left side is positive.

### Conclusion
Since both integer values of \([x]\) lead to no real solutions, we conclude that the original equation \(6x^2 - 77[x] + 147 = 0\) has **no real solutions**.

### Final Answer
The equation has **0 real solutions**.

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