For the quadratic equation 2x^2 + 4x + k = 0 to have real roots, what must be th

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Question: For the quadratic equation 2x^2 + 4x + k = 0 to have real roots, what must be the condition on k? (2019)

Options:

Correct Answer: k <= 4

Exam Year: 2019

Solution:

The discriminant must be non-negative: 4^2 - 4*2*k >= 0, which simplifies to k <= 4.

For the quadratic equation 2x^2 + 4x + k = 0 to have real roots, what must be the condition on k? (2019)

For the quadratic equation 2x^2 + 4x + k = 0 to have real roots, what must be the condition on k? (2019)

Step 1: Identify the quadratic equation, which is 2x^2 + 4x + k = 0.
Step 2: Recall that for a quadratic equation ax^2 + bx + c = 0, the discriminant is given by the formula D = b^2 - 4ac.
Step 3: In our equation, a = 2, b = 4, and c = k.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = 4^2 - 4*2*k.
Step 5: Calculate 4^2, which is 16, so we have D = 16 - 8k.
Step 6: For the quadratic equation to have real roots, the discriminant must be non-negative: D >= 0.
Step 7: Set up the inequality: 16 - 8k >= 0.
Step 8: Rearrange the inequality to isolate k: 16 >= 8k.
Step 9: Divide both sides by 8: 2 >= k.
Step 10: This means k must be less than or equal to 4: k <= 4.

Discriminant of a Quadratic Equation – The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac. For the equation to have real roots, the discriminant must be non-negative (D >= 0).
Conditions for Real Roots – Understanding that the condition for a quadratic equation to have real roots is based on the value of the discriminant.