Introduction to Quadratic Equations

This general equation of a quadratic function has the form:

where a,b, and c are constants, and x and y are variables; this means the values of a,b, and c are chosen, and then kept fixed. The quadratic equation can then be plotted on a x-y graph.

The presence of the x squared term is what gives quadratic equations their fascinating properties, now if a=0 the quadratic equation would reduced to a linear equation. We first study the simplest quadratic equation where a=1, b=0, and c=0:

The x variable can be though of as the length of the sides of a square, and the y variable is then the area of the square. The plot of the area y against x produces are curve which has parabolic shape. The x-values has been extended to negative values (shown in green). A negative dimension for a square has no physical meaning, but mathematically the area y is still positive since the square of a negative number is a positive value. The graph also has a line of mirror symmetry which is the line x=0.

The simplest linear equation is y=x; here a=0, b=1, and c=0. This term can be thought of as defining a line with length y; this keeps with the analogy that the x squared term defines a rectangle.

The constant term y=c where c is a fixed number. The graph of y=c is a horizontal line where the y-coordinate of the line is the value of c.

A boy went shopping for his mother who was too busy. In the first shop he bought 8 apples and 3 oranges, and in the second shop he bought 3 apples and 4 oranges. As soon as he got home his mother asked him "so what have you bought?"; and there can only be one answer 11 apples and 7 oranges. The apples and oranges are "like terms" and like terms can be added together. There is no meaning in adding oranges to apples since these are unlike terms (unless you are making a smoothie) but let's not complicate the discussion. An equation may contain several x squared terms, linear terms, and numbers; and all of these are like terms which can collected togther.

Simplify the expression:

Simplify the expression below by collecting like terms:

Check your answer below:

The equation below shows two brackets side by side, and even though there is no multiplication sign (blue letters) the brackets are multiplied together. The multiplication sign (red equation) in virtually all textbook is not shown since the reader is expected to know that the brackets are multiplied together.

The expansion of the brackets can be performed by apply an algorithm where terms from the first bracket are multiplied with terms in the second bracket. The algorithm can be understood by relating the equation to a rectangle.

The brackets in the above equation can represent the sides of a rectangle; accordingly y is then the area of the rectangle as shown:

The rectangle is divided into four smaller rectangles. The area of each of the small rectangles are found in turn and added together to find the area of the large rectangle. The equation is then simplified by collecting like terms.

Checking your algebraic steps

Substitute any number for x (apart from 0 and 1) into the original equation and calculate y, then expand the brackets and substitute the same value of x to calculate y; obtaining the same value of y suggests that the expansion and simplification has been done correctly.

Expand the brackets and simplify:

Summary

(1) Like terms must be collected when simplifying an equation with terms that can have different powers of x.

(2) The expansion of the product of two linear brackets can be understood by thinking of each bracket as a side of a rectangle, and then finding a simplfied expression for the area of the rectangle.

(3) Algebraic steps can be checked by substituting a value for x (except 0 and 1) and calculating y. The y value will not change if the algebra has been performed correctly.