Lesson 100: Solving Quadratic Equations by Graphing

New Concept

The solutions of the equation are called roots and can be found by determining the $x$-intercepts or zeros of the quadratic function.

What’s next

Next, you’ll learn the step-by-step process to graph these parabolas and pinpoint the exact roots of the equations.

Property

The solutions of a quadratic equation, $0 = ax^2 + bx + c$, are its roots. These can be found by graphing the related function, $f(x) = ax^2 + bx + c$, and identifying the x-intercepts, also known as the zeros of the function, where the U-shaped parabola crosses the x-axis.

Explanation

Think of it as a treasure hunt where 'X' marks the spot! The solutions, or roots, are the treasures. You find them by drawing the parabola and seeing exactly where it crosses the horizontal x-axis. Each crossing point is a solution to the original equation. It's a cool visual way to see the answers instead of just calculating them!

Examples

To solve $x^2 - 16 = 0$, graph the function $f(x) = x^2 - 16$. The graph crosses the x-axis at $x=4$ and $x=-4$, so those are the solutions.
The equation $x^2 - 8x + 16 = 0$ has a related function $f(x) = x^2 - 8x + 16$. Its graph touches the x-axis at a single point, $x=4$, which is the solution.

Let's turn an equation into a picture to find its hidden solutions. This first example shows us how to solve a quadratic equation by graphing the related function.

Example Problem

Solve the equation $x^2 - 16 = 0$ by graphing the related function.

Property

A quadratic graph reveals the number of real solutions:

• Two Real Solutions: The parabola intersects the x-axis at two different points.
• One Real Solution: The parabola's vertex touches the x-axis at a single point.
• No Real Solution: The parabola does not intersect the x-axis at all.

Explanation

Imagine a dolphin jumping out of the water! If it enters and exits the water, it hits the surface (the x-axis) twice, giving two solutions. If it just taps the surface with its tail, that's one solution. If it jumps but never touches the surface at all, there is no real solution. The graph tells the whole story!

Examples

The graph of $f(x) = x^2 - 9$ crosses the x-axis twice, giving two real solutions: $x=3$ and $x=-3$.
The graph of $f(x) = (x-5)^2$ only touches the x-axis at its vertex, $(5, 0)$, so it has one real solution: $x=5$.
The graph of $f(x) = x^2 + 2$ is entirely above the x-axis and never crosses it, so it has no real solutions.

What if the equation isn't ready to be graphed? Let's see how a quick rewrite reveals a simple solution. This example focuses on another key idea, which is that we need to write the equation in standard form first.

Example Problem

Solve the equation $x^2 + 9 = 6x$ by graphing.

Property

When the coefficient of the $x^2$-term in $f(x) = ax^2 + bx + c$ is positive ($a > 0$), the parabola opens upward. When the coefficient is negative ($a < 0$), the parabola opens downward.

Explanation

It's all about attitude! A positive leading coefficient is like a smiley face, so the parabola opens up in a U-shape. A negative leading coefficient is a frowny face, so the parabola opens downward in an upside-down U. Just check the sign of the very first term, and you'll know which way your graph is headed immediately!

Examples

The function $f(x) = 2x^2 - 4x + 1$ opens upward because the coefficient of $x^2$ is a positive 2.
The function $f(x) = -3x^2 + 5x + 7$ opens downward because the coefficient of $x^2$ is a negative 3.