Given \( p+q = 10 \) and \( p^2 + q^2 = 58 \), we first find \( pq \) using:
["Understanding Quadratic Relationships: Solving for ( pq ) When Given ( p+q = 10 ) and ( p^2 + q^2 = 58 )", "When faced with the equations:", "[\np + q = 10 \quad \ ext{and} \quad p^2 + q^2 = 58,\n]", "mathematicians and students alike often seek to find the product ( pq ). While direct computation might seem elusive, leveraging algebraic identities provides a clear and elegant path forward.", "---", "### The Key Identity: ( (p + q)^2 = p^2 + 2pq + q^2 )", "We start with the well-known expansion:", "[\n(p + q)^2 = p^2 + 2pq + q^2.\n]\nThis identity links the sum, the sum of squares, and the product of ( p ) and ( q ).", "Given:\n[\np + q = 10 \quad \Rightarrow \quad (p + q)^2 = 10^2 = 100.\n]\nAnd:\n[\np^2 + q^2 = 58.\n]", "Substituting into the identity:\n[\n100 = 58 + 2pq.\n]", "Solving for ( pq ):\n[\n2pq = 100 - 58 = 42 \quad \Rightarrow \quad pq = \frac{42}{2} = 21.\n]", "---", "### Why This Matters", "The product ( pq ) plays a central role in constructing quadratic equations where ( p ) and ( q ) are roots. For instance, a quadratic equation with roots ( p ) and ( q ) is:", "[\nx^2 - (p+q)x + pq = 0 \quad \Rightarrow \quad x^2 - 10x + 21 = 0.\n]", "Factoring confirms the roots:\n[\n(x - 3)(x - 7) = 0 \quad \Rightarrow \quad x = 3 \ ext{ or } x = 7,\n]\nwhich satisfies both ( p + q = 10 ) and ( p^2 + q^2 = 9 + 49 = 58 ).", "---", "### Conclusion", "Given ( p+q = 10 ) and ( p^2 + q^2 = 58 ), the product ( pq ) is found efficiently using the identity:", "[\np^2 + q^2 = (p + q)^2 - 2pq.\n]", "This approach transforms an abstract problem into a straightforward algebraic solution—ideal for solving quadratic relationships in algebra and beyond. Whether verifying roots or exploring quadratic forms, mastering this identity empowers deeper mathematical insight."]
