What number needs to be added to both sides of the equation x² + 3x = 6 to complete the square?

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To complete the square for the quadratic equation x² + 3x = 6, we need to transform the left-hand side into a perfect square trinomial.

Here are the steps to determine what number should be added:

  1. Start with the quadratic expression on the left side:
    • x² + 3x

2. To complete the square, we need to find the value of (b/2)², where b is the coefficient of x. In our case, b = 3. So:

  • Calculate b/2: 3/2 = 1.5
  • Now, square it: (1.5)² = 2.25

Now we can add this number to both sides of the equation:

  1. Our original equation is:
    • x² + 3x = 6
  2. Add 2.25 to both sides:
    • x² + 3x + 2.25 = 6 + 2.25
  3. This simplifies to:
    • x² + 3x + 2.25 = 8.25

3. The left side can now be factored as:

  • (x + 1.5)² = 8.25

In conclusion, the number that should be added to both sides of the equation to complete the square is 2.25.

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