Lesson 1: Solving Quadratic Equations Using Graphs and Tables

Property

A quadratic equation is an equation that can be written in the standard form:

where $a$, $b$, and $c$ are real numbers and $a \neq 0$.

Examples

• $2x^2 + 5x - 3 = 0$ is a quadratic equation where $a=2$, $b=5$, and $c=-3$.
• $x^2 = 16$ is a quadratic equation because it can be rewritten as $x^2 - 16 = 0$.
• $5x^2 - 2x = 0$ is a quadratic equation where $c=0$.

Explanation

A quadratic equation is a second-degree polynomial equation, meaning the highest exponent of the variable is 2. The term $ax^2$ is the quadratic term, $bx$ is the linear term, and $c$ is the constant term. The condition that $a \neq 0$ is crucial, as the equation would become linear if $a$ were zero. Understanding this standard form is the first step to analyzing and solving these types of equations.

Property

The zeros of a function $f(x)$ are the values of $x$ for which $f(x) = 0$. These values are also the solutions, or roots, of the equation $f(x) = 0$.
For a quadratic function $f(x) = ax^2 + bx + c$, the zeros are the solutions to the quadratic equation $ax^2 + bx + c = 0$.

Examples

• If $x = 3$ is a zero of the function $f(x) = x^2 - 9$, then $f(3) = 3^2 - 9 = 9 - 9 = 0$.
• The zeros of the function $g(x) = x^2 - 5x + 6$ are $x = 2$ and $x = 3$, because $g(2)=0$ and $g(3)=0$. These are the solutions to the equation $x^2 - 5x + 6 = 0$.
• On a graph, the zeros of a function are the x-coordinates of the points where the graph intersects the x-axis, also known as the x-intercepts.

Explanation

A "zero" of a function is an input value that results in an output of zero. Finding the zeros of a quadratic function is the same as solving the related quadratic equation. Graphically, the real zeros of a function correspond to the x-intercepts of its graph. This is because any point on the x-axis has a y-coordinate of zero.

Property

To find the $x$-intercepts of the graph of

we set $y = 0$ and solve the equation