What constant term should be used to complete the square for the expression x² + 5x – 7?

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To complete the square for the expression x² + 5x – 7, we first focus on the quadratic and linear terms, which are x² and 5x. The goal is to transform this part of the expression into a perfect square trinomial.

Here are the steps:

  1. Take the coefficient of the x term (which is 5), divide it by 2, and then square the result. This can be expressed mathematically as follows:

    (5 / 2)² = 25 / 4

  2. Add and subtract this constant term (25/4) inside the expression:

    x² + 5x + (25 / 4) – (25 / 4) – 7

    This rearrangement allows us to express the original quadratic expression in a way that highlights the perfect square.

  3. Next, the expression can be rewritten as:

    (x + 5/2)² – 25/4 – 7

  4. Now, to simplify further, convert -7 into a fraction with the same denominator:

    -7 = -28/4

  5. This gives us:

    (x + 5/2)² – 25/4 – 28/4 = (x + 5/2)² – 53/4

So, the constant term that should be added to complete the square in the expression x² + 5x – 7 is 25/4.

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