The square of 9 less than a number is 3 less than the number. What is the number?

We are given the problem: 

"The square of 9 less than a number is 3 less than the number."

Let's solve this step by step.

1. Let the unknown number be \( x \).

2. The phrase "9 less than a number" means \( x - 9 \).

3. The square of 9 less than a number is \( (x - 9)^2 \).

4. The phrase "is 3 less than the number" means \( x - 3 \).

Now we can set up the equation:

\[
(x - 9)^2 = x - 3
\]

### Solving the equation:

1. Expand the left-hand side of the equation:
\[
(x - 9)^2 = x^2 - 18x + 81
\]

2. The equation becomes:
\[
x^2 - 18x + 81 = x - 3
\]

3. Move all terms to one side of the equation:
\[
x^2 - 18x + 81 - x + 3 = 0
\]

Simplifying further:
\[
x^2 - 19x + 84 = 0
\]

4. Solve this quadratic equation using the quadratic formula:

The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where:
- \( a = 1 \)
- \( b = -19 \)
- \( c = 84 \)

Substitute these values:
\[
x = \frac{-(-19) \pm \sqrt{(-19)^2 - 4(1)(84)}}{2(1)}
\]
\[
x = \frac{19 \pm \sqrt{361 - 336}}{2}
\]
\[
x = \frac{19 \pm \sqrt{25}}{2}
\]
\[
x = \frac{19 \pm 5}{2}
\]

So, the two possible solutions are:
\[
x = \frac{19 + 5}{2} = \frac{24}{2} = 12
\]
or
\[
x = \frac{19 - 5}{2} = \frac{14}{2} = 7
\]

### Checking both solutions:

1. If \( x = 12 \):
\[
(x - 9)^2 = (12 - 9)^2 = 3^2 = 9
\]
and
\[
x - 3 = 12 - 3 = 9
\]
Both sides are equal, so \( x = 12 \) is a solution.

2. If \( x = 7 \):
\[
(x - 9)^2 = (7 - 9)^2 = (-2)^2 = 4
\]
and
\[
x - 3 = 7 - 3 = 4
\]
Both sides are equal, so \( x = 7 \) is also a solution.

### Final Answer:
The number can be either \( 12 \) or \( 7 \).

Would you like more details on any step?

Here are 5 related questions:
1. What are the different methods to solve quadratic equations?
2. Can we solve this problem using a graph?
3. How can we derive the quadratic formula?
4. What are some common mistakes when solving quadratic equations?
5. How do you determine if a quadratic equation has real or complex solutions?

**Tip:** Always check both possible solutions when using the quadratic formula to ensure none are extraneous.

Learn how to solve the problem where the square of 9 less than a number equals 3 less than the number. This step-by-step guide covers quadratic equations, the quadratic formula, and algebraic problem solving. Suitable for students in Grades 8-10.

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