The average of a positive number, its square, and 3 is 5. Find the number.

Question

Accepted Answer

#### Step 1: Formulate the Equation
Let the positive number be $$ x $$. The average of $$ x $$, $$ x^2 $$, and 3 is given as 5.  
The equation for the average is:  
$$
\frac{x + x^2 + 3}{3} = 5
$$

#### Step 2: Simplify the Equation
Multiply through by 3 to eliminate the denominator:  
$$
x + x^2 + 3 = 15
$$  
Rearrange terms:  
$$
x^2 + x - 12 = 0
$$

#### Step 3: Solve the Quadratic Equation
The quadratic equation is:  
$$
x^2 + x - 12 = 0
$$  
Factorize or use the quadratic formula. Factoring gives:  
$$
(x - 3)(x + 4) = 0
$$  
Thus, the solutions are:  
$$
x = 3 \quad 	ext{or} \quad x = -4
$$

#### Step 4: Select the Positive Solution
Since $$ x $$ must be positive, we choose $$ x = 3 $$.

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Thus, the number is **3**, making Option **(b)** correct.

Answer

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