When a number is subtracted from its square results 20

Let's solve the problem step by step.

Let the number be \( x \).

According to the problem, when the number is subtracted from its square, the result is 20. This can be written as:

\[
x^2 - x = 20
\]

Now, rearrange the equation:

\[
x^2 - x - 20 = 0
\]

This is a quadratic equation. We can solve it using the quadratic formula:

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

For the equation \( x^2 - x - 20 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -1 \)
- \( c = -20 \)

Substitute these values into the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-20)}}{2(1)}
\]
\[
x = \frac{1 \pm \sqrt{1 + 80}}{2}
\]
\[
x = \frac{1 \pm \sqrt{81}}{2}
\]
\[
x = \frac{1 \pm 9}{2}
\]

This gives us two possible solutions:

\[
x = \frac{1 + 9}{2} = \frac{10}{2} = 5
\]
\[
x = \frac{1 - 9}{2} = \frac{-8}{2} = -4
\]

So, the number can be either \( 5 \) or \( -4 \).

Would you like more details or have any questions?

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Here are five related questions:

1. How can you verify if the solutions are correct?
2. What is the difference between completing the square and using the quadratic formula?
3. How do you derive the quadratic formula?
4. Can all quadratic equations be solved using factoring?
5. What are the roots of a quadratic equation, and how are they related to its graph?

**Tip:** Quadratic equations can have two, one, or no real solutions, depending on the value of the discriminant (\( b^2 - 4ac \)).

Learn how to solve the problem where a number, when subtracted from its square, results in 20. This problem involves solving a quadratic equation using the quadratic formula, covering key algebraic concepts and methods. Suitable for students in Grades 9-12.

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