efere10.17. 1) \( \frac{5 x^{2}+1}{2}=\frac{1+7 x}{4}+\frac{1+8 x}{9} \)

Let's simplify the equation step by step! First, we start with the original equation: \[ \frac{5 x^{2}+1}{2}=\frac{1+7 x}{4}+\frac{1+8 x}{9} \] To eliminate the fractions, find a common denominator for the fractions on the right side. The least common multiple of 4 and 9 is 36. So, we will multiply the entire equation by 36: \[ 36 \cdot \frac{5 x^{2}+1}{2} = 36 \cdot \left( \frac{1 + 7x}{4} + \frac{1 + 8x}{9} \right) \] Now, simplifying both sides yields: \[ 18(5x^2 + 1) = 9(1 + 7x) + 4(1 + 8x) \] Expanding both sides: \[ 90x^2 + 18 = 9 + 63x + 4 + 32x \] \[ 90x^2 + 18 = 13 + 95x \] Next, we rearrange everything to one side of the equation: \[ 90x^2 - 95x + 5 = 0 \] Now, we can use the quadratic formula where \( a = 90 \), \( b = -95 \), and \( c = 5 \): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substituting in our values: \[ x = \frac{-(-95) \pm \sqrt{(-95)^2 - 4 \cdot 90 \cdot 5}}{2 \cdot 90} \] \[ x = \frac{95 \pm \sqrt{9025 - 1800}}{180} \] \[ x = \frac{95 \pm \sqrt{7225}}{180} \] \[ x = \frac{95 \pm 85}{180} \] This results in two possible solutions: 1. \( x = \frac{180}{180} = 1 \) 2. \( x = \frac{10}{180} = \frac{1}{18} \) Thus, the solutions to the given equation are \( x = 1 \) and \( x = \frac{1}{18} \).