Why Quadratics?

A number of students seem to find the introduction of quadratic equations frustrating. After spending much time learning about linear equations, and finally just getting to the point where everything seems to be starting to make sense and be “easy” again, all of a sudden the teacher starts in on a totally different and seemingly unrelated topic…

However, quadratics are not a totally different topic. Linear equations (where the highest power of a variable is the first power) and quadratic equations (where the highest power of a variable is the second power) are the least complex two members of the family of polynomial equations.

y=2x+4~~~~~~~~~~~~~~~~~\text{linear equation, or 1st degree polynomial}\\*~\\*y=3x^2-7x+1~~~~~~~~\text{quadratic equation, or 2nd degree polynomial}\\*~\\*y=2x^3-5x^2+x-7~\text{cubic equation, or 3rd degree polynomial}\\*~\\*\text{etc.}

The algebra, graphing, and analysis skills learned in solving and graphing polynomials will be useful in many math and science courses that follow. They are also needed in many quantitative professions such as business, medicine, construction, etc.

Quadratic expressions arise in problems involving areas (quad is a Latin prefix for four, as in the four sides of a rectangle). Areas are measured in square units (square feet, square miles), and are the result of multiplying two variables together. Notice that when two of the same variable are multiplied together, the result is a variable to the second power, a “squared” quantity (x^2 is the area of a square with sides of length x).

Quadratic expressions arise in product packaging problems (quantity of material needed to hold a product), coverage problems (seeds to cover a field, or paint to cover a wall), and problems that involve constant acceleration, such as motion that is affected only by gravity (a pebble flying through the air, if wind resistance is ignored).

Solving linear equations requires the ability to “undo” each of the first four operations learned in elementary school: addition, subtraction, multiplication, and division. Note that addition and subtraction are inverse operations (they undo one another), and the same goes for multiplication and division. By combining like terms, moving all terms with variables to one side of the equation, and/or moving all constant terms to the other side, you can simplify a linear equation to the point where one final step (dividing both sides by the coefficient of the variable) will produce the value being sought.

Manipulating quadratic expressions (no equal sign) and solving quadratic equations (expressions on both sides of an equal sign) requires you to bring the last two operations learned into play: exponents and roots. They will also give you a great deal more experience with both distributing and factoring (inverse processes), and recognizing or creating “perfect squares” (both as numbers and as algebraic expressions) – because square roots only produce “nice” answers when applied to perfect squares.

In short, your work with quadratics will both re-enforce your current algebra skills and expand them to a new level. You will also learn more about how to sketch the graph of a function, recognize one function as a translation and/or dilation of another, and solve systems of non-linear equations (equations that describe graphical shapes other than lines). And lastly, you will gain the math skills needed to answer questions like “if I throw a stone straight up with a speed of 10 ft per second, how long will the stone be in the air before it falls back to the height from which I threw it?” Or, “How high will the stone get before it begins to fall back to earth?”.

Being able to figure out the answers to such questions by yourself is pretty neat!

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Published by

Whit Ford

Math Tutor in Yarmouth, Maine

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