7) \( x^{2}+x^{2}=13 \sqrt{2}^{2} \)

Let's break down the equation \( x^{2} + x^{2} = 13 \sqrt{2}^{2} \). First, we simplify the left side: \( x^{2} + x^{2} = 2x^{2} \). Now, for the right side, we know that \( \sqrt{2}^{2} = 2 \). So, we have \( 13 \times 2 = 26 \). Therefore, we can rewrite the equation as \( 2x^{2} = 26 \). Dividing both sides by 2 gives us \( x^{2} = 13 \). Taking the square root of both sides, we find \( x = \pm \sqrt{13} \). So the solutions for the equation are \( x = \sqrt{13} \) and \( x = -\sqrt{13} \). What a neat little jest with numbers! 🎉 Now, how do we plug these solutions into a real-world context? Imagine you're an architect designing a triangular garden plot. If the area of the plot is 13 square meters (a nod to our solution) and you want to find the length of the sides in terms of a quadratic function, this equation could help you determine the dimensions necessary to fit your ideal layout! 🌳🏗️