Setting Up the Equation

Let the positive number be represented by the variable $x$. According to the problem statement, when this number is subtracted from its square ($x^2$), the result is 812.

This can be written as the equation:

$x^2 - x = 812$

Solving the Quadratic Equation

To find the value of $x$, we first rearrange the equation into the standard quadratic form $ax^2 + bx + c = 0$:

$x^2 - x - 812 = 0$

We can solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

In this equation, $a=1$, $b=-1$, and $c=-812$.

First, calculate the discriminant ($\Delta = b^2 - 4ac$):

$\Delta = (-1)^2 - 4(1)(-812)$

$\Delta = 1 + 3248$

$\Delta = 3249$

Now, find the square root of the discriminant:

$\sqrt{3249} = 57$

Now, apply the quadratic formula to find the possible values for $x$:

$x = \frac{-(-1) \pm 57}{2(1)}$

$x = \frac{1 \pm 57}{2}$

This gives two possible solutions:

• $x_1 = \frac{1 + 57}{2} = \frac{58}{2} = 29$
• $x_2 = \frac{1 - 57}{2} = \frac{-56}{2} = -28$

Finding the Positive Number

The problem asks for a positive number. Therefore, we choose the positive solution.

The positive number is $x = 29$.

Verification

Let's check the answer:

Square of the number: $29^2 = 841$.

Subtract the number from its square: $841 - 29 = 812$.

The result matches the condition given in the problem.