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