1. Stating the problem: Solve the system: $$25x^2=16$$ $$\frac{x}{x+2}=\frac{x+3}{5x+11}$$ 2. Solve the first equation: $$25x^2=16 \implies x^2=\frac{16}{25} \implies x=\pm \frac{4}{5}$$ 3. Solve the second equation: Start with: $$\frac{x}{x+2}=\frac{x+3}{5x+11}$$ Cross multiply: $$x(5x+11) = (x+3)(x+2)$$ Expand both sides: $$5x^2 + 11x = x^2 + 5x + 6$$ Bring all terms to one side: $$5x^2 + 11x - x^2 - 5x - 6 = 0$$ Simplify: $$4x^2 + 6x - 6 = 0$$ Divide entire equation by 2: $$2x^2 + 3x - 3 = 0$$ 4. Solve quadratic: Use quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ with $a=2$, $b=3$, $c=-3$ Calculate discriminant: $$\Delta = 3^2 - 4 \cdot 2 \cdot (-3) = 9 + 24 = 33$$ Compute solutions: $$x= \frac{-3 \pm \sqrt{33}}{4}$$ Final answers: From first equation: $$x = \pm \frac{4}{5}$$ From second equation: $$x = \frac{-3 + \sqrt{33}}{4}$$ or $$x = \frac{-3 - \sqrt{33}}{4}$$