Question

Which expression is equivalent to negative x squared plus 14 x minus 49?

Question 1 Answer options with 4 options
A.
negative left-parenthesis x plus 7 right-parenthesis left-parenthesis x minus 7 right-parenthesis

B.
negative left-parenthesis x minus 7 right-parenthesis left-parenthesis x minus 7 right-parenthesis

C.
left-parenthesis negative x plus 7 right-parenthesis left-parenthesis x plus 7 right-parenthesis

D.
left-parenthesis negative x minus 7 right-parenthesis left-parenthesis negative x minus 7 right-parenthesis

Question 2
,begin emphasis,Part B,end emphasis,
How many zeroes, if any, does the expression negative x squared plus 14 x minus 49 have?

Question 2 Answer options with 4 options
A.
no zeroes

B.
one zero

C.
two zeroes

D.
three zeroes

Answer 1

To solve the first question, we will factor the expression \(-x^2 + 14x - 49\).

1. First, we can factor out a negative sign:
\[-(x^2 - 14x + 49)\]

2. Next, we will factor the quadratic \(x^2 - 14x + 49\). We need to find two numbers that multiply to \(49\) (the constant term) and add to \(-14\) (the coefficient of \(x\)). These numbers are \(-7\) and \(-7\), so:

\[x^2 - 14x + 49 = (x - 7)(x - 7) = (x - 7)^2\]

3. Thus, we rewrite the expression:
\[-(x - 7)^2\]

Now we can match this with the answer options:
- A. \(- (x + 7)(x - 7)\)
- B. \(- (x - 7)(x - 7)\) (This matches)
- C. \(( -x + 7)(x + 7)\)
- D. \(( -x - 7)( -x - 7)\)

So the answer for Question 1 is **B**: \(- (x - 7)(x - 7)\).

For Question 2, we will determine how many zeroes the expression has. The factored form we found is:

\[-(x - 7)^2 = 0\]

Setting this equal to 0:
\[(x - 7)^2 = 0\]

This equation has one solution:
\[x - 7 = 0 \implies x = 7\]

Therefore, the expression has **one zero**.

So, the answer for Question 2 is **B**: one zero.

To summarize:
- **Question 1**: B
- **Question 2**: B