Learn on PengiCalifornia Reveal Math, Algebra 1Unit 10: Quadratic Functions

10-3 Solving Quadratic Equations by Graphing

In this Grade 9 lesson from California Reveal Math, Algebra 1 (Unit 10), students learn to solve quadratic equations by graphing the related quadratic function and identifying its x-intercepts, also called zeros or roots. The lesson covers equations with two solutions, one solution, and no real solutions, connecting the standard form ax² + bx + c = 0 to the behavior of its parabola. Students also practice approximating solutions using tables when x-intercepts fall between integer values.

Section 1

Rewriting a Quadratic Equation in Standard Form

Property

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

ax^2 + bx + c = 0, \quad a \neq 0

Section 2

Solving Quadratic Equations by Graphing

Property

To Solve a Quadratic Equation by Graphing:

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

Section 3

Number of x-intercepts

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:

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

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

Section 4

Estimating Solutions Using Sign Changes in Tables

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

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Rewriting a Quadratic Equation in Standard Form

Property

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

ax^2 + bx + c = 0, \quad a \neq 0

Section 2

Solving Quadratic Equations by Graphing

Property

To Solve a Quadratic Equation by Graphing:

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

Section 3

Number of x-intercepts

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:

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

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

Section 4

Estimating Solutions Using Sign Changes in Tables

Property

Examples

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Rewriting a Quadratic Equation in Standard Form

Property

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

ax^2 + bx + c = 0, \quad a \neq 0

Section 2

Solving Quadratic Equations by Graphing

Property

To Solve a Quadratic Equation by Graphing:

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

Section 3

Number of x-intercepts

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:

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

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

Section 4

Estimating Solutions Using Sign Changes in Tables