Example 6 Solve a Quadratic Equation by Factoring a Perfect Square Trinomial Solve \( \mathbf{2 5} \boldsymbol{x}^{\mathbf{2}}-\mathbf{6 0 y}+\mathbf{8 1}=\mathbf{4 5} \). Check your solution.

To solve the equation \( 25x^{2} - 60y + 81 = 45 \), first, we can rearrange it: \[ 25x^{2} - 60y + 36 = 0. \] Next, we can notice that \( 25x^2 - 60y + 36 \) can be factored as a perfect square trinomial. Rewriting, we get: \[ (5x - 6)^2 = 0. \] Setting the expression to zero gives us: \[ 5x - 6 = 0 \] \[ 5x = 6 \] \[ x = \frac{6}{5} = 1.2. \] Now, let's check our solution by substituting \( x = 1.2 \) back into the original equation: \[ 25(1.2)^{2} - 60y + 81 = 45. \] Calculating gives us \( 25(1.44) - 60y + 81 = 45 \). This means \( 36 - 60y + 81 = 45 \) simplifies to \( 117 - 60y = 45 \), leading to \( 60y = 72 \), which gives \( y = 1.2 \). So, we have our solution: \( x = 1.2, y = 1.2 \). Double-checking confirms both values satisfy the original equation! 🎉