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Write h(x) = 7 + 10x + x2 in vertex form.



Write h in standard form.
h(x) = x2 + 10x + 7
Form a perfect square trinomial by adding and subtracting StartFraction b Over 2 EndFraction squared.
h(x) = (x2 + 10x + 25) + 7 – 25
Write the trinomial as a binomial squared.
Write the function in vertex form, if needed.
What is h(x) = 7 + 10x + x2 written in vertex form?

h(x) = (x – 25)2 – 18
h(x) = (x – 5)2 + 32
h(x) = (x + 5)2 – 18
h(x) = (x + 25)2 + 32

1 answer

To convert the function \( h(x) = 7 + 10x + x^2 \) into vertex form, we'll follow the steps you outlined.

First, we'll start with the standard form:

\[ h(x) = x^2 + 10x + 7 \]

Next, to form a perfect square trinomial, we will complete the square. The term for completing the square is calculated by taking half of the coefficient of \( x \) (which is 10) and squaring it:

\[ \left(\frac{10}{2}\right)^2 = 5^2 = 25 \]

Now we can rewrite the function by adding and subtracting 25:

\[ h(x) = (x^2 + 10x + 25) + 7 - 25 \]

This simplifies to:

\[ h(x) = (x + 5)^2 - 18 \]

So, the vertex form of the function is:

\[ h(x) = (x + 5)^2 - 18 \]

Now, let's evaluate the options provided:

  1. \( h(x) = (x - 25)^2 - 18 \) (incorrect)
  2. \( h(x) = (x - 5)^2 + 32 \) (incorrect)
  3. \( h(x) = (x + 5)^2 - 18 \) (correct)
  4. \( h(x) = (x + 25)^2 + 32 \) (incorrect)

Therefore, the correct answer is:

\[ h(x) = (x + 5)^2 - 18 \]