Let \( a \) and \( b \) be complex numbers such that \( a + b = 4 + 2i \) and \( ab = 5 + 3i \). Find the value of \( a^2 + b^2 \).
["Finding ( a^2 + b^2 ) Given Complex Numbers ( a ) and ( b )", "When working with complex numbers in algebraic problems, one powerful identity helps simplify expressions involving sums and products:", "[\na^2 + b^2 = (a + b)^2 - 2ab\n]", "This identity is especially useful when the sum ( a + b ) and the product ( ab ) are known—exactly the case here.", "We are given:\n- ( a + b = 4 + 2i )\n- ( ab = 5 + 3i )", "Let’s compute ( a^2 + b^2 ) step by step using the identity.", "Step 1: Compute ( (a + b)^2 )\n[\n(a + b)^2 = (4 + 2i)^2 = 4^2 + 2 \cdot 4 \cdot 2i + (2i)^2 = 16 + 16i + 4i^2\n]\nSince ( i^2 = -1 ),\n[\n16 + 16i + 4(-1) = 16 + 16i - 4 = 12 + 16i\n]", "Step 2: Compute ( 2ab )\n[\n2ab = 2(5 + 3i) = 10 + 6i\n]", "Step 3: Apply the identity\n[\na^2 + b^2 = (a + b)^2 - 2ab = (12 + 16i) - (10 + 6i) = (12 - 10) + (16i - 6i) = 2 + 10i\n]", "Thus, the value of ( a^2 + b^2 ) is\n[\n\boxed{2 + 10i}\n]", "This algebraic technique efficiently connects sum and product of complex numbers to their squared sums—often used in solving quadratic equations with complex roots and in signal processing or quantum mechanics applications.", "Keywords: complex numbers, ( a^2 + b^2 ), complex algebra, ( a + b = 4 + 2i ), ( ab = 5 + 3i ), identity ( a^2 + b^2 = (a + b)^2 - 2ab ), solving complex equations, complex identities."]
