Find two numbers with a product of 44 and a sum of 45.
There are many ways of approaching this problem, where we need to find two numbers that multiply to give 44 and add to give 45. We could simply use trial and error or trial and improvement. However, in this question, we will use an algebraic method and begin by letting the two numbers be 𝑥 and 𝑦. This gives us two equations: 𝑥𝑦 is equal to 44 and 𝑥 plus 𝑦 equals 45.
We have a pair of simultaneous equations that we can solve by substitution. Subtracting 𝑥 from both sides of the second equation gives us 𝑦 is equal to 45 minus 𝑥. We can now substitute this expression for 𝑦 into equation one. This gives us 𝑥 multiplied by 45 minus 𝑥 is equal to 44. Distributing the parentheses or expanding the brackets gives us 45𝑥 minus 𝑥 squared. And this is equal to 44. Next, we can add 𝑥 squared and subtract 45𝑥 from both sides. This gives us zero is equal to 𝑥 squared minus 45𝑥 plus 44.
We now have a quadratic that we can solve by factoring. Since negative one multiplied by negative 44 gives us 44 and negative one plus negative 44 is negative 45, our quadratic can be factored to 𝑥 minus one multiplied by 𝑥 minus 44. Since the product of 𝑥 minus one and 𝑥 minus 44 equals zero, then either 𝑥 minus one equals zero or 𝑥 minus 44 equals zero. Adding one to both sides of our first equation gives us 𝑥 equals one. And adding 44 to both sides of the second equation gives us 𝑥 equals 44.
Since 𝑦 is equal to 45 minus 𝑥, the corresponding values of 𝑦 are 44 and one. This means that the only pair of numbers that have a product of 44 and a sum of 45 are one and 44.