Thrice the square of a natural number decreased by four times the number is equal to 50 more than the number. The number is:

Options

A. 4
B. 5
C. 10
D. 6

Answer

To solve the problem step by step, let's first translate the given statement into a mathematical equation.

### Step 1: Define the variable
Let the natural number be represented by \( x \).

### Step 2: Translate the problem into an equation
According to the problem, we have:
- Thrice the square of the number: \( 3x^2 \)
- Four times the number: \( 4x \)
- 50 more than the number: \( x + 50 \)

The equation can be set up as:
\[ 3x^2 - 4x = x + 50 \]

### Step 3: Rearrange the equation
To solve for \( x \), we need to rearrange the equation:
\[ 3x^2 - 4x - x - 50 = 0 \]
This simplifies to:
\[ 3x^2 - 5x - 50 = 0 \]

### Step 4: Solve the quadratic equation
Now we will use the quadratic formula to solve for \( x \):
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 3 \), \( b = -5 \), and \( c = -50 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot (-50) \]
\[ = 25 + 600 \]
\[ = 625 \]

Now substituting back into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{625}}{2 \cdot 3} \]
\[ = \frac{5 \pm 25}{6} \]

### Step 5: Calculate the possible values of \( x \)
Calculating the two possible values:
1. \( x = \frac{30}{6} = 5 \)
2. \( x = \frac{-20}{6} = -\frac{10}{3} \) (not a natural number)

Since \( x \) must be a natural number, we take:
\[ x = 5 \]

### Conclusion
The natural number is \( 5 \).

---