The image shows two geometric shapes, a square and a right triangle, with given side lengths in terms of $x$.

Information from the image:

1. Square: The side length of the square is labeled as $3x + 2$ cm.
2. Right Triangle: One leg is $4$ cm, and the hypotenuse is labeled as $4x + 8$ cm.

Solution:

To find $x$, we need to apply the Pythagorean Theorem to the triangle.

The formula is: $\text{(Leg)}^2 + \text{(Other Leg)}^2 = \text{Hypotenuse}^2$

In this triangle, we have:

• One leg: $4$ cm (given).
• Hypotenuse: $4x + 8$ cm.

Since it appears we only have one leg and the hypotenuse, the other leg is likely the same as the side of the square, $3x + 2$.

Now, using the Pythagorean theorem: $4^2 + (3x + 2)^2 = (4x + 8)^2$

Expanding each term:

1. $4^2 = 16$
2. $(3x + 2)^2 = 9x^2 + 12x + 4$
3. $(4x + 8)^2 = 16x^2 + 64x + 64$

Substitute and simplify: $16 + 9x^2 + 12x + 4 = 16x^2 + 64x + 64$ $9x^2 + 12x + 20 = 16x^2 + 64x + 64$

Move all terms to one side: $0 = 16x^2 + 64x + 64 - 9x^2 - 12x - 20$ $0 = 7x^2 + 52x + 44$

Now, solve the quadratic equation $7x^2 + 52x + 44 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 7$, $b = 52$, and $c = 44$.

Calculating: $x = \frac{-52 \pm \sqrt{(52)^2 - 4 \cdot 7 \cdot 44}}{2 \cdot 7}$ $x = \frac{-52 \pm \sqrt{2704 - 1232}}{14}$ $x = \frac{-52 \pm \sqrt{1472}}{14}$

Simplify further to find the possible values of $x$.

Would you like me to continue with the calculation for $x$, or do you have any questions about the process so far?

Related Questions

1. How do you solve quadratic equations using the quadratic formula?
2. What is the Pythagorean Theorem, and how is it applied in solving problems with right triangles?
3. What steps are necessary for expanding and simplifying polynomial expressions?
4. How do you determine if a solution is physically meaningful in geometry problems?
5. How would the solution change if the side lengths were defined differently?

Tip: In quadratic equations, double-check your constants and terms for accuracy to avoid calculation errors.