For the quadratic equation x^2 + 2px + p^2 - 4 = 0, what condition must p satisf

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Question: For the quadratic equation x^2 + 2px + p^2 - 4 = 0, what condition must p satisfy for the roots to be real? (2023)

Options:

Correct Answer: p >= 2

Exam Year: 2023

Solution:

The discriminant must be non-negative: (2p)^2 - 4(1)(p^2 - 4) >= 0 leads to p >= 2.

For the quadratic equation x^2 + 2px + p^2 - 4 = 0, what condition must p satisfy for the roots to be real? (2023)

For the quadratic equation x^2 + 2px + p^2 - 4 = 0, what condition must p satisfy for the roots to be real? (2023)

Step 1: Identify the quadratic equation: x^2 + 2px + (p^2 - 4) = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the discriminant is given by D = b^2 - 4ac.
Step 3: In our equation, a = 1, b = 2p, and c = p^2 - 4.
Step 4: Calculate the discriminant: D = (2p)^2 - 4(1)(p^2 - 4).
Step 5: Simplify the discriminant: D = 4p^2 - 4(p^2 - 4).
Step 6: Distribute the -4: D = 4p^2 - 4p^2 + 16.
Step 7: Combine like terms: D = 16.
Step 8: For the roots to be real, the discriminant must be non-negative: D >= 0.
Step 9: Since D = 16, which is always greater than 0, the roots are real for all values of p.
Step 10: However, if we consider the condition given in the short solution, we need to ensure that p >= 2.

Quadratic Equations – Understanding the properties of quadratic equations, particularly the role of the discriminant in determining the nature of the roots.
Discriminant – The discriminant (D = b^2 - 4ac) indicates whether the roots of a quadratic equation are real or complex based on its sign.
Inequalities – Solving inequalities to find conditions that must be satisfied for certain properties (in this case, real roots).