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## Quadratic Equation Worksheets

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What is a quadratic equation?

A quadratic equation is an equation that has a single variable squared. So, x2 = 10 is a quadratic equation. The normal way to write this same equation would be x2 - 10 = 0. Quadratics can have other terms, but they have to have a squared variable. The general form of the quadratic equation is: ax2 + bx + c = 0, but b and c can be zero, and the a can be one.

A quadratic equation is a type of polynomial. It is sometimes called a polynomial in the second degree, because the first x has an exponent of 2.

 How are quadratic equations used in the real world? An equation that describes motion is often quadratic, as are equations that describe the shape of things like satellite dishes (parabolic reflectors) and lenses. One example can be seen on suspension bridges. The main cables are in the shape of a parabola and the formula for them is quadratic. Galileo described the trajectory of cannon balls as quadratic, with the maximum range being 1/16 times v2 (where v is the speed at which the cannon ball comes out of the cannon. Until his analysis, artillerymen didn't know exactly how their cannon balls flew, and aiming was a best-guess estimate. Even today artillery shots are calculated with the science of ballistics and use the same equations of motion. They also calculate other variables, like the drag an object feels when it travels through the air, how wind and air density affect a trajectory and other factors - but the underlying equations of motion haven't changed. A basic problem with quadratic equations. When polynomials are multiplied together, a quadratic equation often is the result. Multiplying (x + 10) and (x + 10) equals x2 + 20x + 100. Setting this to equal zero allows us to find x = -10 when x2 + 20x + 100 = 0. Finding the zero tells us where a graph of y = x2 + 20x + 100 touches the X axis. It gives us information about what this particular curve looks like. This is especially important when you want to find the lowest quantity that y can be. Who invented quadratic equations? Although these types of equations have been known as early as the Babylonians (around 1800 BC) they weren't really addressed as a class until around 700 BC in India. Geometric methods of solving quadratics appear in a religious text on altar construction that dates from that time. The quadratic formula, which allows solution of any quadratic equation, didn't appear in modern form until almost 1900. The formula looks like this: it uses the same letters as the general formula, ax2 + bx + c = 0 and gives two values for x (the two values may be the same). The symbol ± means addition (for one answer) and subtraction (to get the other answer). An interesting fact about quadratic equations. Knowing how quadratic equations come about is sometimes useful for 'lightning calculators". Lightning calculators are people who can do complicated mathematics in their heads. There are many techniques used to accomplish such speed math. For instance, suppose you are asked to multiply 98 and 102 in your head. Knowing quadratic equations allows you to think of this as (100 - 2) times (100 + 2). This helps because for any number, (x + a) X (x - a) = x2 - a2. In our example we think of 98 and 102 as 1002 - 22 = 10,000 - 4 = 9,996. This way of using the quadratic will give a quick answer whenever the square of the middle number is known. If you know that 20 squared is 400, you can almost immediately answer the question, "What is 23 times 17?" by just using the same pattern.