Quadratics Explained: Easy Steps for MYP Students

Welcome to our exploration of quadratic equations!A quadratic equation is a special type of equation where the highest power of x is 2.Let's break down the standard form. We have a, b, and c as numbers that determine the equation's behavior.To understand quadratics visually, let's look at their graphs.The simplest quadratic equation is x squared. When we graph it, it forms a U-shaped curve called a parabola.If we make the coefficient of x squared negative, the parabola flips upside down.A more complex quadratic equation like x squared plus three x plus two still forms a parabola, but with a different position and shape.Every quadratic equation has some important characteristics. The graph is always a parabola, which is symmetric around its vertex. The direction it opens depends on whether a is positive or negative.To factor this quadratic equation, we'll follow a systematic approach.First, we check for any common factors that can be factored out. In this case, there aren't any.Next, we look for two numbers that multiply to give c, which is 6, and add to give b, which is 5.Let's check each pair. We need numbers that multiply to 6 and add to 5.Two and three work perfectly because they multiply to give 6 and add to give 5.Now we can write our quadratic in factored form.To find the solutions, we use the zero product property. If a product is zero, one of its factors must be zero.Let's try a slightly more complex example with a common factor.Here we can factor out 2 first, then factor the remaining quadratic just like before.One of the most common real-world examples of quadratics is the path of a thrown object, like a basketball.The height of the basketball can be modeled by a quadratic equation. Here, negative 4.9 t squared represents gravity's effect, while the other terms represent initial velocity and height.Another important application is in business, where profit can be modeled as a quadratic function. Let's look at how revenue and costs interact.Revenue typically increases with quantity but levels off due to market saturation, following a curved pattern.Costs often increase more rapidly as production scales up, following another curved pattern.The difference between revenue and costs gives us our profit curve, which is also quadratic.Quadratics also appear in architecture. The cables of a suspension bridge naturally form a parabolic shape.This shape is described by a simple quadratic equation, where the cable's height increases with the square of the horizontal distance from the center.Quadratics appear in many other real-world applications. Here are a few more examples.Understanding quadratics helps us model, predict, and optimize many aspects of our world.Thanks for exploring the real-world applications of quadratics with Spark.E!