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=(x+3)(x+4)
y equals left parenthesis x plus 3 right parenthesis left parenthesis x plus 4 right parenthesis

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

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

y=(x−4)

1 answer

To determine which equation gives the correct number of real and non-real solutions corresponding to the number of x-intercepts shown on a graph, we need to analyze the potential x-intercepts of each equation provided.

  1. Equation: \( y = (x + 3)(x + 4) \)

    • This equation has two real roots: \( x = -3 \) and \( x = -4 \). Therefore, it has 2 x-intercepts.
  2. Equation: \( y = (x^2 - 2)(x + 4) \)

    • The term \( x^2 - 2 = 0 \) has two real roots: \( x = \sqrt{2} \) and \( x = -\sqrt{2} \). The term \( x + 4 \) has one real root at \( x = -4 \). So, this equation has a total of 3 real roots and 3 x-intercepts.
  3. Equation: \( y = (x^2 + 2)(x + 4) \)

    • The term \( x^2 + 2 = 0 \) has no real roots (it has 2 non-real complex solutions). The term \( x + 4 \) has one real root at \( x = -4 \). So, this equation has 1 real root and 2 non-real roots, resulting in 1 x-intercept.
  4. Equation: \( y = (x - 4) \)

    • This equation has one real root at \( x = 4 \), giving it 1 x-intercept.

Now, depending on the graph and the number of x-intercepts it shows, you can match the equations with how many real and non-real solutions there are.

Summary of x-intercepts for each equation:

  • The first equation has 2 real x-intercepts.
  • The second equation has 3 real x-intercepts.
  • The third equation has 1 real x-intercept and 2 complex solutions.
  • The fourth equation has 1 real x-intercept.

Please refer to the specific x-intercepts shown on the graph to choose the appropriate equation.