How can I solve the equation 2x^2 + 8 = 0 by graphing the related function?

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Solving the Equation 2x² + 8 = 0 by Graphing

To solve the equation 2x² + 8 = 0 by graphing, you first want to rewrite the equation in a form that represents a function. To do this, we can express the equation as:

y = 2x² + 8

Next, we need to plot the graph of this quadratic function. Here are the steps to follow:

1. Identify Key Features of the Function

  • Vertex: The vertex can be calculated using the formula x = -b/(2a), where y = ax² + bx + c is the standard form of a quadratic equation. In our case, since there is no x term, b = 0
  • Axis of Symmetry: The line x = -b/(2a) also serves as the axis of symmetry.

For our quadratic function:

  • a = 2
  • b = 0
  • c = 8

Calculating the vertex gives:
x = -0/(2*2) = 0

Next, plug that back into the function to find the y-coordinate:

y = 2(0)² + 8 = 8

So, the vertex is at (0, 8).

2. Plotting Points

To visualize the graph, we can calculate a few additional points. You might choose values like x = -2, x = -1, x = 1, and x = 2:

  • x = -2: y = 2(-2)² + 8 = 8 + 8 = 16 → Point: (-2, 16)
  • x = -1: y = 2(-1)² + 8 = 2 + 8 = 10 → Point: (-1, 10)
  • x = 1: y = 2(1)² + 8 = 2 + 8 = 10 → Point: (1, 10)
  • x = 2: y = 2(2)² + 8 = 8 + 8 = 16 → Point: (2, 16)

3. Sketch the Graph

Now that we have the vertex and several points, you can plot these points on a Cartesian plane:

1. Mark the vertex at (0, 8).

2. Plot the points (-2, 16), (-1, 10), (1, 10), and (2, 16).

3. Draw a smooth parabola opening upwards through these points. Since all y-values are positive, the graph never intersects the x-axis. This indicates that the equation 2x² + 8 = 0 has no real solutions.

Conclusion

In summary, graphing the function y = 2x² + 8 reveals that it does not cross the x-axis. Therefore, the equation 2x² + 8 = 0 has no real solutions.

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