1. Solve $5r^2 + 10r = 80$ by completing the square. Step 1: Divide the entire equation by 5 to simplify: $$r^2 + 2r = 16$$ Step 2: To complete the square, add and subtract the square of half the coefficient of $r$, which is $1^2 = 1$: $$r^2 + 2r + 1 = 16 + 1$$ Step 3: Rewrite the left side as a perfect square: $$(r + 1)^2 = 17$$ Step 4: Solve for $r$ by taking the square root of both sides: $$r + 1 = \pm \sqrt{17}$$ Step 5: Isolate $r$: $$r = -1 \pm \sqrt{17}$$ --- Verify the solution using the quadratic formula for $5r^2 + 10r - 80 = 0$: Step 1: Identify coefficients: $$a=5, \quad b=10, \quad c=-80$$ Step 2: Use the quadratic formula: $$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-10 \pm \sqrt{100 - 4 \cdot 5 \cdot (-80)}}{10}$$ Step 3: Simplify under the square root: $$\sqrt{100 + 1600} = \sqrt{1700} = \sqrt{100 \cdot 17} = 10 \sqrt{17}$$ Step 4: Plug back: $$r = \frac{-10 \pm 10\sqrt{17}}{10} = -1 \pm \sqrt{17}$$ The quadratic formula matches the completing square result. --- 2. Solve $25x^2 - 49 = 0$ using the quadratic formula and verify with difference of squares. Quadratic formula: Step 1: Identify coefficients: $$a=25, \quad b=0, \quad c=-49$$ Step 2: Use quadratic formula: $$x = \frac{-0 \pm \sqrt{0 - 4 \cdot 25 \cdot (-49)}}{2 \cdot 25} = \frac{\pm \sqrt{4900}}{50}$$ Step 3: Simplify square root: $$\sqrt{4900} = 70$$ Step 4: Calculate solutions: $$x = \frac{\pm 70}{50} = \pm \frac{7}{5}$$ Difference of squares factoring: Step 1: Rewrite left side: $$25x^2 - 49 = (5x)^2 - 7^2$$ Step 2: Factor as difference of squares: $$(5x - 7)(5x + 7) = 0$$ Step 3: Solve each factor: $$5x - 7 = 0 \Rightarrow x = \frac{7}{5}$$ $$5x + 7 = 0 \Rightarrow x = -\frac{7}{5}$$ Both methods give the same answer $x = \pm \frac{7}{5}$.