Solving Systems of Quadratic Equations

A quadratic equation occurs when you have one or more terms containing a variable raised to the second power. For example, the equation y = Sx2 + 2x is a quadratic equation. Solving systems of equations involving quadratic equations leads to some interesting results. You usually end up with more than one solution to the system. Sometimes more than one solution means that there s more than one answer. Often, though, it means that one of the answers fits the situation and the other is just extraneous — it satisfies the system of equations, but it doesn t really have any meaning in the practical problem. 

A common type of problem involving systems of equations is a number problem, where you solve for a certain number that has a particular relationship to one or more other numbers. You incorporate operations such as addition and multiplication, and you often use squares of numbers or their reciprocals. 

The Problem The sum of two numbers is 40, and the product of those same two numbers is 204. What are the numbers  

Let the two numbers be represented by the variables x and y Writing that their sum is 40, the equation is x + y = 40. You write that their product is 204 with xy = 204. Solve for y in the first equation to get y = 40 - x Now replace the y in the second equation with 40 - x, simplify the equation by setting it equal to 0, and solve the quadratic equation by factoring. 

You get that x is either 34 or 6. These are the two numbers whose sum is 40 and whose product is 204. 

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