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

Using discriminants to graph parabolas -
Professor Edward Burger explains using discriminants to graph parabolas

Graphing some important functions - Ploting points. linear functions, quadratic functions, cubic functions, square root functions, absolute value functions
Professor Edward Burger explains graphing some important functions

Stretching a graph - Parabola shapes and cubic shapes
Professor Edward Burger explains stretching a graph.

Graphing quadratics using patterns - summary
Professor Edward Burger explains graphing quadratics using patterns.

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