We know \( a + b = 10 \) and \( a^2 + b^2 = 58 \). Use the identity:

["Understanding the Symbolic Equation: ( a + b = 10 ) and ( a^2 + b^2 = 58 ) with the Power Identity", "When given the equations ( a + b = 10 ) and ( a^2 + b^2 = 58 ), many students feel puzzled—until they apply a key mathematical identity that reveals deeper insights. This powerful tool helps simplify expressions and uncover hidden relationships between variables. In this article, we’ll explore how to use the identity involving the square of a sum to solve for individual and combined values of ( a ) and ( b ), and why it’s essential for algebra and problem-solving.", "---", "### The Core Identity: ( (a + b)^2 = a^2 + b^2 + 2ab )", "At the heart of this problem lies a fundamental identity in algebra:", "[\n(a + b)^2 = a^2 + b^2 + 2ab\n]", "This identity connects the sum of two numbers and their squares directly to their product. Knowing this helps us bridge unknowns when only partial information is provided. Given that ( a + b = 10 ) and ( a^2 + b^2 = 58 ), we can use this identity to find ( ab )—a crucial stepping stone to solving for ( a ) and ( b ).", "---", "### Step 1: Square the Sum ( (a + b)^2 )", "Start by squaring both sides of ( a + b = 10 ):\n[\n(a + b)^2 = 10^2 = 100\n]", "---", "### Step 2: Substitute Known Values into the Identity", "Plug ( a^2 + b^2 = 58 ) into the identity:\n[\n100 = 58 + 2ab\n]", "Now solve for ( 2ab ):\n[\n2ab = 100 - 58 = 42\n]", "---", "### Step 3: Solve for ( ab )", "Divide by 2:\n[\nab = \frac{42}{2} = 21\n]", "Now we know:\n- ( a + b = 10 )\n- ( ab = 21 )", "These two values are enough to characterize ( a ) and ( b ) completely, since they are the roots of the quadratic equation:\n[\nx^2 - (a + b)x + ab = 0 \quad \Rightarrow \quad x^2 - 10x + 21 = 0\n]", "---", "### Step 4: Solve the Quadratic Equation", "Factor the equation:\n[\nx^2 - 10x + 21 = (x - 3)(x - 7) = 0\n]", "Thus, the solutions are:\n[\na = 3, \quad b = 7 \quad \ ext{or} \quad a = 7, \quad b = 3\n]", "Order doesn’t matter in this context—both satisfy the original equations.", "---", "### Why This Identity Matters: Real-World and Academic Applications", "Understanding the relationship between sums and squares is not just theoretical—it’s foundational in:\n- Data analysis: Computing variance and standard deviation from summed values.\n- Physics: Relating total energy (sum of kinetic and potential) to individual energy components.\n- Algebra: Simplifying expressions and solving systems of equations efficiently.", "This identity transforms abstract equations into solvable systems by leveraging algebraic structure—a skill essential for advanced mathematics and STEM fields.", "---", "### Conclusion", "Given just ( a + b = 10 ) and ( a^2 + b^2 = 58 ), we used the identity ( (a + b)^2 = a^2 + b^2 + 2ab ) to uncover ( ab = 21 ), then derived the individual values. This approach exemplifies how powerful algebraic identities can turn limited information into complete understanding. Whether for homework help, standardized tests, or deeper study, mastering this technique unlocks powerful problem-solving capabilities.", "Remember:\nWhen faced with ( a + b ) and ( a^2 + b^2 ), leverage the identity:\n[\n(a + b)^2 = a^2 + b^2 + 2ab\n]\nto find ( ab ), then factor or solve to reveal ( a ) and ( b ).", "Happy solving!", "---", "Keywords: ( a + b = 10 ), ( a^2 + b^2 = 58 ), algebraic identity, solve for variables, understand equations, algebra identity, mathematical proof, quadratic equation, factoring, sum and squares identity.", "Meta Description:\nDiscover how to use ( (a + b)^2 = a^2 + b^2 + 2ab ) to solve for ( a ) and ( b ) when ( a + b = 10 ) and ( a^2 + b^2 = 58 ). Learn step-by-step algebra techniques for instant problem solving."]