10-3 Solving Quadratic Equations by Graphing

Property

A quadratic equation is in standard form when it is written as:

Property

To Solve a Quadratic Equation by Graphing:

1. Graph the quadratic function $y = ax^2 + bx + c$
2. Find where the parabola crosses the $x$-axis (the $x$-intercepts)
3. The $x$-coordinates of these intersection points are the solutions to the equation $ax^2 + bx + c = 0$

Property

The $x$-intercepts of the graph of $y = ax^2 + bx + c$ are the solutions of $ax^2 + bx + c = 0$. There are three possibilities:

1. If both solutions are real numbers, and unequal, the graph has two $x$-intercepts.

2. If the solutions are real and equal, the graph has one $x$-intercept, which is also its vertex.

Property

When a quadratic function $f(x) = ax^2 + bx + c$ changes from positive to negative (or negative to positive) between consecutive x-values in a table, a zero exists between those x-values.
The solution can be estimated by creating additional tables with smaller intervals around the sign change.

Examples