Algebra Tutorials!
Wednesday 28th of June
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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Solving Quadratic Equations by Graphing

Objective Understand why the solutions of a quadratic equation occur where the graph of the corresponding function intersects the x-axis.

Let's see some examples, where the goal is to find or approximate the solutions to a quadratic equation ax 2 + bx + c = 0 by looking to see where the graph of the corresponding quadratic equation intersects the x-axis.

Example 1

Solve x 2 - x - 2 = 0 by graphing.

Solution

Step 1 Make a table of values, and then graph the function y = x 2 - x - 2.

Step 2 Find where the parabola intersects the x -axis. In this case, this occurs when x = -1 and x = 2. Thus, - 1 and 2 are the solutions to the equation x 2 - x - 2 = 0 .

Step 3 Check the answer by factoring.

0 = x 2 - x - 2

0 = ( x + 1)( x - 2)

Set each factor equal to zero.

 x + 1 = 0 x - 2 = 0 x = -1 x + 2

So, the solutions check.

Example 2

Solve x 2 - 2x - 2 = 0 by graphing.

Solution

Step 1 Make a table of values, and then graph the function y = x 2 - 2x - 2.

Step 2 Notice that the parabola does not intersect the x-axis. This parabola opens upward and we can compute that the vertex is at (1, 1). This vertex lies above the x-axis, so the parabola never intersects the x-axis. We can conclude that there are no solutions to the equation x 2 - 2x - 2 = 0.

Example 3

Solve by graphing.

Solution

Step 1 Make a table of values, and then graph the function .

Step 2 Try to factor and solve for the roots. In this case, does not have integral factors.

Step 3 Use the graph to approximate the solutions. We can see from the graph that the parabola intersects the x -axis twice: once between -3 and -2, and once roughly halfway between 0 and 1. We can therefore estimate solutions to the equation to be . (In this case, these two estimates are actually exact solutions.)

In general, graphs provide a good way of approximating solutions to quadratic equations when the corresponding quadratic expressions cannot be factored with integral factors. In order to get exact solutions, we need to use the Quadratic Formula, which will be discussed in other lessons.