Graphical Solutions Of Quadratic Equations

Example:

Solve the equation x² + x – 3 = 0 by drawing its graph for –3 ≤ x ≤ 2.

Solution:

Draw the graph for y =x² + x – 3 for –3 ≤ x ≤ 2.

The solution for the equation x² + x – 3 can be obtained by looking at the points where the graph y = x² + x – 3 cuts the x-axis (i.e. y = 0).

The graph y = x² + x – 3, cuts the x-axis at x 1.3 and x –2.3

So, the solution for the equation x + x –3 is x 1.3 or x –2.3.

Recall that in the quadratic formula, the discriminant b² – 4ac is positive when there are two distinct real roots.

Example:

By plotting the graph, solve the equation 6x – 9 – x² = 0.

Solution:

Notice that the graph does not cross the x-axis, but touches the x-axis at x = 3. This means that the equation 6x – 9 – x² = 0 has equal roots of x = 3.

Recall that in the quadratic formula, in such a case where the roots are equal, the discriminant b² – 4ac = 0.

Example:

Solve the equation x² + 4x + 8 = 0 using the graphical method.

Solution:

Notice that the graph does not cross or touch the x-axis. This means that the equation x² + 4x + 8 = 0 does not have any real roots.

Recall that in the quadratic formula, the discriminant b² – 4ac, is negative when there are no real roots.

Videos

Solving Quadratic Equations by Graphing Part 1
This video demonstrates how to solve quadratic equations by graphing.
First, a quadratic equation is converted into a quadratic function.
Then, the variables are changed to x and y to graph on a coordinate plane. The solutions are shown where the function crosses the x-axis.
Roots, x-intercepts, and zeros are given as synonyms for solutions. Finding roots from a table of values is also demonstrated.

Solving Quadratic Equations by Graphing Part 2
This video shows how to solve quadratic equations using the TI84 and TI83 series of graphing calculators.
Five problems are worked out. The different steps are shown including converting quadratic equations into calculator ready graphable quadratic functions.
The video shows how to examine in graph and table view what the solutions are. The case of having no solutions is shown as well as that of having only one solution.

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