The sum of two numbers is 20, and their product is 96. What is the larger number?
["Sum and Product of Two Numbers: Finding the Larger Number When Sum is 20 and Product is 96", "Have you ever come across a classic math puzzle where the sum of two numbers is 20 and their product is 96? If you’re eager to solve it quickly and understand the logic, you’re in the right place.", "This type of problem is a perfect example of using algebra to find unknown values based on two key pieces of information:\n- The sum of the numbers: ( a + b = 20 )\n- Their product: ( a \ imes b = 96 )", "Step-by-Step Solution", "1. Set up the equations:\n Let the two numbers be ( a ) and ( b ).\n [\n a + b = 20 \ ag{1}\n ]\n [\n a \ imes b = 96 \ ag{2}\n ]", "2. Express one variable in terms of the other:\n From equation (1), solve for ( b ):\n [\n b = 20 - a\n ]", "3. Substitute into the product equation:\n Replace ( b ) in equation (2):\n [\n a \ imes (20 - a) = 96\n ]\n Expand:\n [\n 20a - a^2 = 96\n ]", "4. Rearrange into standard quadratic form:\n [\n -a^2 + 20a - 96 = 0\n ]\n Multiply through by -1 to simplify:\n [\n a^2 - 20a + 96 = 0\n ]", "5. Solve the quadratic equation:\n Use the quadratic formula:\n [\n a = \frac{20 \pm \sqrt{(-20)^2 - 4 \cdot 1 \cdot 96}}{2 \cdot 1} = \frac{20 \pm \sqrt{400 - 384}}{2} = \frac{20 \pm \sqrt{16}}{2}\n ]\n [\n a = \frac{20 \pm 4}{2}\n ]", "So:\n [\n a = \frac{24}{2} = 12 \quad \ ext{or} \quad a = \frac{16}{2} = 8\n ]", "6. Find the larger number:\n If ( a = 12 ), then ( b = 20 - 12 = 8 )\n If ( a = 8 ), then ( b = 20 - 8 = 12 )\n The larger number is clearly 12.", "Conclusion:\nThe sum of two numbers is 20, and their product is 96. By solving using algebra, the larger number is 12. This method applies to many algebraic word problems and strengthens problem-solving skills applicable in math competitions, school lessons, and real-world applications.", "---", "Keyword-rich takeaway for SEO:\nSum of two numbers is 20, product is 96, solve for larger number, algebra problem, quadratic equations, math puzzle solution. Use these insights to master similar problems and boost confidence in solving sum and product questions."]
