Question

Which of the following equations would give the correct number of real and non-real solutions to match the number of x-intercepts shown on the graph?

(1 point)
Responses

y=(x2+2)(x+4)
𝑦
=
(
𝑥
2
+
2
)
(
𝑥
+
4
)
y equals left parenthesis x squared plus 2 right parenthesis left parenthesis x plus 4 right parenthesis

y=(x−4)
𝑦
=
(
𝑥
−
4
)
y equals left parenthesis x minus 4 right parenthesis

y=(x2−2)(x+4)
𝑦
=
(
𝑥
2
−
2
)
(
𝑥
+
4
)
y equals left parenthesis x squared minus 2 right parenthesis left parenthesis x plus 4 right parenthesis

y=(x+3)(x+4)
𝑦
=
(
𝑥
+
3
)
(
𝑥
+
4
)

Answer 1

To determine which equation corresponds to the number of x-intercepts shown on the graph, we need to analyze the possible x-intercepts based on the factorizations provided in each option.

1. **y = (x² + 2)(x + 4)**:
- The factor \( x² + 2 \) does not produce real x-intercepts because it is always positive for all real \( x \) (as \( x² + 2 \) has no real roots). The factor \( x + 4 \) gives 1 real root (x-intercept at \( x = -4 \)). Therefore, this function has 1 real x-intercept and all other roots are non-real.

2. **y = (x − 4)**:
- This equation has 1 real x-intercept at \( x = 4 \) (1 real solution and no non-real solutions).

3. **y = (x² − 2)(x + 4)**:
- The factor \( x² - 2 \) has 2 real roots (x-intercepts at \( x = ±\sqrt{2} \)), and the factor \( x + 4 \) has 1 real root (x-intercept at \( x = -4 \)). Therefore, this function has a total of 3 real x-intercepts and no non-real solutions.

4. **y = (x + 3)(x + 4)**:
- Both factors provide real solutions (x-intercepts at \( x = -3 \) and \( x = -4 \)). Therefore, this function has 2 real x-intercepts and no non-real solutions.

**Conclusion:**
- The correct choice depends on how many x-intercepts are shown on the graph you are comparing these equations against. If you need to match the number of x-intercepts derived from the equations to what is shown on the graph, you would select the equation that has the same number of real solutions as the x-intercepts displayed.

- If there are no x-intercepts, then the first option is the right match (y = (x² + 2)(x + 4)).
- If there is 1 x-intercept, then choose the second option (y = (x − 4)).
- If there are 2 x-intercepts, then choose the fourth option (y = (x + 3)(x + 4)).
- If there are 3 x-intercepts, then choose the third option (y = (x² − 2)(x + 4)).

Please confirm how many x-intercepts are shown on your graph for a precise answer!