Basics of Quadratic Equations

What are quadratic equations

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Let's Take a Look First

Understanding comes quicker when we first look at things that can be easily visualized before thinking about them. While quadratic equations and quadratic functions are different, we can visually understand their properties by graphing the quadratic function $y=ax^2+bx+c$ corresponding to the quadratic equation $ax^2+bx+c=0$ .

Graphing a Quadratic Function

Let's start by drawing the graph of the simplest quadratic function $y=ax^2$ . Press the Start button on the graph below to plot points on $y=ax^2$ . (We're changing the x-coordinate of point $P$ from $-3$ to $3$ in increments of $0.1$ , within the domain $-3 \leqq x \leqq 3$ .)

While the graph of a linear function $y=ax+b$ was a straight line, the graph of a quadratic function is a parabola . As its name suggests, a parabola is a curve that has the same shape as the trajectory of an object thrown at an angle.

Now, let's look at the properties of quadratic equations while keeping this shape in mind.

Properties of Quadratic Equations

First, let's review the prerequisite knowledge necessary to understand the properties of quadratic equations.

Review of Prerequisites

Equations

An equation is an equality that contains a variable as an unknown. An equality is an expression where two numbers or expressions are connected by an equals sign (=), and the left-hand side and right-hand side have equal values. We solve for $x$ by transforming the equation into the form $x=...$ using methods like transposition while maintaining this equality relationship.

Since a quadratic equation is also an equation, we can find the value of $x$ by solving for $x$ .

Degree

In 7th grade, we learned about linear equations. The difference between linear equations and quadratic equations is their degree . For example, when solving an equation for $x$ , the degree of $x$ in a linear equation is 1, but in a quadratic equation, the degree of $x$ is 2 ( $x^2$ ).

Square Root

A square root is a number $x$ that, when squared, becomes $a$ . The square root of $a$ is the value of $x$ that satisfies $x^2=a$ . Among the square roots of $a$ , we denote the positive one as $\sqrt{a}$ and call it "root $a$ ".

Solutions of Quadratic Equations

Let's consider the quadratic equation $x^2=4$ , for example. The meaning of this equation is what is $x$ when squared becomes 4 . Since 4 is the square of $\pm\,2$ , we get the solution $x=\pm\,2$ . (Don't forget that square roots exist as both positive and negative numbers!)

$$\begin{aligned} x^2 &= 4 \\ x &= \pm\,2 \end{aligned}$$

The fact that the degree of $x$ is 2 means that there can be two solutions for $x$ . While linear equations had one solution, quadratic equations have two solutions. (There are also cases where there's only one solution or no solution, which we'll deal with later.)

Graph of the Function Corresponding to a Quadratic Equation

Now, let's graph the quadratic function $y=x^2-4$ corresponding to this quadratic equation $x^2=4$ . Where do the solutions $x=\pm\,2$ appear on the graph?

Also, use the dropdown in the upper left corner of the graph area to check the graphs of quadratic functions corresponding to quadratic equations like $x^2=9$ and $x^2=16$ .

The solutions to the quadratic equation $ax^2+bx+c=0$ are the values that satisfy $y=0$ when considering the quadratic function, so the intersections of the parabola graph with the x-axis are the solutions. Since the shape of the graph is a parabola, you can imagine that it intersects the x-axis at two points (there are two solutions).