Understanding the Problem: Finding the Positive Number

The question asks us to find a positive number based on a relationship between its value, its square, and 21 times its value. The relationship is given as: twenty-one times the number is less than its square by 100.

Translating the Word Problem into an Equation

Let's represent the positive number we are trying to find with the variable \(x\). Since the number must be positive, we know that \(x > 0\).

According to the problem statement:

• "Twenty-one times of a positive number" can be written as \(21x\).
• "its square" can be written as \(x^2\).
• "is less than ... by 100" means the difference between the larger quantity (\(x^2\)) and the smaller quantity (\(21x\)) is 100.

So, we can write the equation:

\(x^2 - 21x = 100\)

To solve this, we need to rearrange it into the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\). Subtract 100 from both sides:

\(x^2 - 21x - 100 = 0\)

Here, we have a quadratic equation with \(a=1\), \(b=-21\), and \(c=-100\).

Solving the Quadratic Equation to Find the Value

We can solve this quadratic equation using factoring. We need to find two numbers that multiply to \(c = -100\) and add up to \(b = -21\).

Let's list some pairs of factors for -100:

• 1 and -100 (Sum = -99)
• -1 and 100 (Sum = 99)
• 2 and -50 (Sum = -48)
• -2 and 50 (Sum = 48)
• 4 and -25 (Sum = -21)
• -4 and 25 (Sum = 21)
• 5 and -20 (Sum = -15)
• -5 and 20 (Sum = 15)
• 10 and -10 (Sum = 0)

The pair 4 and -25 satisfies both conditions: \(4 \times (-25) = -100\) and \(4 + (-25) = -21\). So, we can factor the quadratic equation as:

\((x + 4)(x - 25) = 0\)

For this equation to be true, one of the factors must be zero.

• Case 1: \(x + 4 = 0\)
• Solving for x: \(x = -4\)

• Case 2: \(x - 25 = 0\)
• Solving for x: \(x = 25\)

Considering the Positive Number Constraint

The question specifically asks for a positive number. From our solutions, we have \(x = -4\) and \(x = 25\). Since the number must be positive (\(x > 0\)), the solution \(x = -4\) is not valid for this problem.

Therefore, the only valid solution is \(x = 25\).

Verification of the Positive Number Value

Let's check if our positive number, 25, satisfies the original condition:

• Twenty-one times the number: \(21 \times 25 = 525\)
• The square of the number: \(25^2 = 625\)
• Is 525 less than 625 by 100? \(625 - 525 = 100\). Yes, it is.

The value 25 satisfies the condition given in the problem.

Conclusion

The positive number that satisfies the given condition is 25.

Revision Table: Positive Number Problem

Additional Information: Solving Quadratic Equations

Quadratic equations of the form \(ax^2 + bx + c = 0\) can be solved using several methods:

• Factoring: This method is used when the quadratic expression can be factored into two linear expressions, like \((px+q)(rx+s) = 0\). The solutions are then \(x = -q/p\) and \(x = -s/r\). This is often the quickest method if factoring is straightforward.
• Completing the Square: This method involves rearranging the equation so that one side is a perfect square trinomial. It can be used for any quadratic equation.
• Quadratic Formula: This formula provides the solutions directly: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It can be used for any quadratic equation and is particularly useful when factoring is difficult or impossible with integer coefficients. The term \(b^2 - 4ac\) is called the discriminant, which tells us about the nature of the solutions (real or complex, distinct or repeated).

In our problem, factoring was a suitable and efficient method to find the positive number.