Solution: Given $ m + n = 10 $, $ m^2 + n^2 = 58 $.
["Solution: How to Solve $ m + n = 10 $ and $ m^2 + n^2 = 58 $ – A Step-by-Step Approach", "When presented with a system involving two equations:", "$$\nm + n = 10 \quad \ ext{(1)}\n$$\n$$\nm^2 + n^2 = 58 \quad \ ext{(2)}\n$$", "many students wonder how to find the values of $ m $ and $ n $ efficiently. This comprehensive guide walks you through solving the system using algebraic identities and provides clear explanations to strengthen your confidence in handling quadratic relationships.", "---", "### Step 1: Recall Key Algebraic Identity", "To connect the sum $ m + n $ with the sum of squares $ m^2 + n^2 $, use the identity:", "$$\n(m + n)^2 = m^2 + 2mn + n^2\n$$", "This identity allows us to express $ m^2 + n^2 $ in terms of $ m + n $ and $ mn $, which is essential for solving the system.", "---", "### Step 2: Plug in Known Values", "From equation (1), we know:", "$$\nm + n = 10 \Rightarrow (m + n)^2 = 10^2 = 100\n$$", "Using the identity:", "$$\n(m + n)^2 = m^2 + n^2 + 2mn\n$$", "Substitute known values:", "$$\n100 = 58 + 2mn\n$$", "---", "### Step 3: Solve for $ mn $", "Subtract 58 from both sides:", "$$\n100 - 58 = 2mn \Rightarrow 42 = 2mn \Rightarrow mn = 21\n$$", "Now we know both the sum and the product of $ m $ and $ n $, meaning $ m $ and $ n $ are roots of the quadratic equation:", "$$\nx^2 - (m + n)x + mn = 0\n\Rightarrow x^2 - 10x + 21 = 0\n$$", "---", "### Step 4: Solve the Quadratic Equation", "Factor the quadratic:", "$$\nx^2 - 10x + 21 = (x - 3)(x - 7) = 0\n$$", "Thus, the solutions are:", "$$\nx = 3 \quad \ ext{or} \quad x = 7\n$$", "Hence, $ m = 3, n = 7 $ or $ m = 7, n = 3 $.", "---", "### Final Thoughts", "Given $ m + n = 10 $ and $ m^2 + n^2 = 58 $, the solution is elegantly derived using algebraic identities and basic algebra. This method is fast and reliable, and it highlights how understanding key formulas enhances problem-solving in algebra.", "Whether you're a student tackling satisfying equations or someone improving mathematical literacy, mastering this approach unlocks stronger analytical skills for more complex problems.", "---", "Summary:\n- Sum: $ m + n = 10 $\n- Sum of squares: $ m^2 + n^2 = 58 $\n- Identity: $ (m + n)^2 = m^2 + n^2 + 2mn $\n- Solved: $ m = 3, n = 7 $ or $ m = 7, n = 3 $\n- Key takeaway: Use identities to connect sum and sum of squares efficiently", "---", "Keywords:\nsolve $ m + n = 10 $, $ m^2 + n^2 = 58 $, algebra solution, quadratic equations, sum and product of roots, algebraic identities, step-by-step method."]
