How do you find the x-intercept of the graph represented by the equation y = x² + 4x + 4?

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To find the x-intercept of the graph of the equation y = x² + 4x + 4, we need to determine the points where the graph intersects the x-axis. This occurs when y = 0.

Here are the steps to find the x-intercept:

  1. Set the equation to zero: Replace y by 0 in the equation:
  2. 0 = x² + 4x + 4
  3. Simplify the equation: Rearranging gives:
  4. x² + 4x + 4 = 0
  5. Factoring the quadratic: The next step is to factor the quadratic expression. The equation can be factored as follows:
  6. (x + 2)(x + 2) = 0 or (x + 2)² = 0
  7. Finding the solutions: Now, set each factor equal to zero:
  8. x + 2 = 0
  9. Calculating: Solving this gives:
  10. x = -2
  11. Conclusion: Therefore, the x-intercept of the graph is at the point (-2, 0). This means that the graph crosses the x-axis at x = -2.

Additionally, it is worth noting that this quadratic equation represents a parabola that opens upwards (since the coefficient of x² is positive), and since it has a double root at x = -2, this is also the vertex of the parabola.

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