The difference of a positive number in its square is 380 find the number

Let's denote the positive number by \( x \). According to the problem, the difference between the number and its square is 380. Mathematically, this can be written as:

\[
x^2 - x = 380
\]

This equation is a quadratic equation. We can rewrite it as:

\[
x^2 - x - 380 = 0
\]

To solve for \( x \), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = -1 \), and \( c = -380 \). Plugging in these values, we get:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-380)}}{2(1)}
\]

Simplifying further:

\[
x = \frac{1 \pm \sqrt{1 + 1520}}{2}
\]

\[
x = \frac{1 \pm \sqrt{1521}}{2}
\]

The square root of 1521 is 39, so:

\[
x = \frac{1 \pm 39}{2}
\]

This gives us two possible solutions:

\[
x = \frac{40}{2} = 20 \quad \text{or} \quad x = \frac{-38}{2} = -19
\]

Since the problem asks for the positive number, the solution is:

\[
x = 20
\]

So, the positive number is **20**.

Would you like more details or have any questions?

Here are 8 related questions you might find interesting:

1. How do you derive the quadratic formula?
2. What are the other methods to solve quadratic equations?
3. Can you solve the equation using factoring?
4. How do you know that 20 is the correct solution, not -19?
5. What is the significance of the discriminant in the quadratic formula?
6. How can quadratic equations be applied in real-life scenarios?
7. How would the problem change if the difference was given as a negative number?
8. What is the relationship between the roots of a quadratic equation and its coefficients?

**Tip:** When solving quadratic equations, always check if the equation can be factored before using the quadratic formula, as factoring can be quicker and simpler.

Discover how to find a positive number whose square differs by 380. Step-by-step solution using the quadratic formula provided. Ideal for high school students learning quadratic equations and their applications.

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