Question

Which quadratic function is a perfect square binomial?(1 point) Responses f(x)=x2+6x+8 f left parenthesis x right parenthesis equals x squared plus 6 x plus 8 f(x)=x2+10x+25 f left parenthesis x right parenthesis equals x squared plus 10 x plus 25 f(x)=x2+7x+12 f left parenthesis x right parenthesis equals x squared plus 7 x plus 12 f(x)=x2−9x+18

Answer 1

A quadratic function is a perfect square binomial if it can be expressed in the form \((x + a)^2\), which expands to \(x^2 + 2ax + a^2\).

Let's analyze the given functions:

1. \(f(x) = x^2 + 6x + 8\)
- The number 6 (the coefficient of x) does not satisfy \(2a\) for any integer \(a\) that would make \(8\) (the constant term) equal to \(a^2\) when substituting \(a = 3\).
- Thus, it is **not** a perfect square binomial.

2. \(f(x) = x^2 + 10x + 25\)
- Here \(10 = 2a\) implies \(a = 5\).
- The constant term \(25\) equals \(5^2\).
- Therefore, it can be written as \((x + 5)^2\), which makes it a perfect square binomial.

3. \(f(x) = x^2 + 7x + 12\)
- For this one, \(7 = 2a\) does not result in an integer \(a\).
- Also, \(12\) is not a perfect square from any integer \(a\).
- Thus, it is **not** a perfect square binomial.

4. \(f(x) = x^2 - 9x + 18\)
- In this case, \(-9\) suggests \(a\) to be \(-4.5\) (from \(2a = -9\)), and \(18\) does not fit as \((-4.5)^2\).
- Thus, it is **not** a perfect square binomial.

The only quadratic function from the list that is a perfect square binomial is:

**f(x) = x^2 + 10x + 25** (which can be expressed as \((x + 5)^2\)).