Differences Between Quadratic & Linear Equations

A linear equation in two variables doesn't involve any power higher than one for either variable. It has the general form:

\(Ax + By + C = 0\)

where A, ​B​ and ​C​ are constants. It's possible to simplify this to

\(y = mx + b\text{ where } m = \frac{ −A}{B}\)

and ​b​ is the value of ​y​ when ​x​ = 0. A quadratic equation, on the other hand, involves one of the variables raised to the second power. It has the general form

\(y = ax^2 + bx + c\)

Apart from the adding complexity of solving a quadratic equation compared to a linear one, the two equations produce different types of graphs.

A linear equation produces a straight line when you graph it. Each value of ​x​ produces one and only one value of ​y​, so the relationship between them is said to be one-to-one. When you graph a quadratic equation, you produce a parabola that begins at a single point, called the vertex, and extends upward or downward in the ​y​ direction. The relationship between ​x​ and ​y​ is not one-to-one because for any given value of ​y​ except the ​y​-value of the vertex point, there are two values for ​x​.

Linear equations in standard form (​Ax​ + ​By​ + ​C​ = 0) are easy to convert to convert to slope intercept form (​y​ = ​mx​ +​b​), and in this form, you can immediately identify the slope of the line, which is ​m​, and the point at which the line crosses the ​y​-axis. You can graph the equation easily, because all you need are two points. For example, suppose you have the linear equation

\(y = 12x + 5\)

Choose two values for ​x​, say 1 and 4, and you immediately get the values 17 and 53 for ​y​. Plot the two points (1, 17) and (4, 53), draw a line through them, and you're done.

You can't solve and graph a quadratic equation quite as simply. You can identify a few general characteristics of the parabola by looking at the equation. For example, the sign in front of the ​x​² term tells you whether the parabola opens up (positive) or down (negative). Moreover, the coefficient of the ​x​² term tells you how wide or narrow the parabola is — large coefficients denote wider parabolas.

You can find the the ​x​-intercepts of the parabola by solving the equation for ​y​ = 0 :

\(ax^2 + bx + c = 0\)

and using the quadratic formula

\(x = \frac{ −b ± \sqrt{b^2 − 4ac}}{2a}\)

You can find the vertex of a quadratic equation in the form

\(y = ax^2 + bx + c\)

by using a formula derived by completing the square to convert the equation into a different form. This formula is

\(\frac{−b}{2a}\)

It gives you the ​x​-value of the intercept, which you can plug into the equation to find the ​y​-value.

Knowing the vertex, the direction in which the parabola opens and the ​x​-intercept points gives you enough of an idea of the appearance of the parabola to draw it.