The image shows a right triangle with one leg labeled $x$, the other leg labeled $x + 9$, and the hypotenuse labeled $45$. The goal is to find the value of $x$.

To solve for $x$, we will use the Pythagorean Theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs. This can be written as:

$c^2 = a^2 + b^2$

Here:

• The hypotenuse $c = 45$,
• One leg $a = x$,
• The other leg $b = x + 9$.

Using the Pythagorean Theorem:

$45^2 = x^2 + (x + 9)^2$

First, we calculate $45^2$:

$2025 = x^2 + (x + 9)^2$

Now, expand $(x + 9)^2$:

$(x + 9)^2 = x^2 + 18x + 81$

Substituting back into the equation:

$2025 = x^2 + x^2 + 18x + 81$

Simplifying:

$2025 = 2x^2 + 18x + 81$

Now, subtract 2025 from both sides:

$0 = 2x^2 + 18x + 81 - 2025$

Simplifying further:

$0 = 2x^2 + 18x - 1944$

Now, we can divide the entire equation by 2 to simplify it:

$0 = x^2 + 9x - 972$

This is a quadratic equation, and we can solve for $x$ using the quadratic formula:

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

For the equation $x^2 + 9x - 972 = 0$, $a = 1$, $b = 9$, and $c = -972$. Plugging in these values:

$x = \frac{-9 \pm \sqrt{9^2 - 4(1)(-972)}}{2(1)}$ $x = \frac{-9 \pm \sqrt{81 + 3888}}{2}$ $x = \frac{-9 \pm \sqrt{3969}}{2}$ $x = \frac{-9 \pm 63}{2}$

This gives two possible solutions:

1. $x = \frac{-9 + 63}{2} = \frac{54}{2} = 27$
2. $x = \frac{-9 - 63}{2} = \frac{-72}{2} = -36$

Since $x$ represents a side length, it must be positive. Therefore, the solution is:

$x = 27$

Would you like more details or have any questions? Here are some related questions you might consider:

1. What would the equation look like if the hypotenuse had a different value?
2. How would the Pythagorean Theorem apply if the triangle were isosceles?
3. Can you use trigonometric ratios to solve this problem?
4. How does the quadratic formula apply to solving real-world problems?
5. What are the conditions for a quadratic equation to have real solutions?

Tip: Always check the discriminant ($b^2 - 4ac$) in the quadratic formula to ensure the nature of the roots (real or complex).