Complex Numbers and Quadratic Equations

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Q1. What is the solution set for the inequality 2x + 1 ≥ 3?

Solution:

Subtract 1 from both sides: 2x ≥ 2. Then divide by 2: x ≥ 1.

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Q2. What is the solution set for the inequality x^2 - 4 > 0?

Solution:

Factor the inequality: (x - 2)(x + 2) > 0. The solution set is x < -2 or x > 2.

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Q3. What is the vertex of the parabola represented by the equation y = x^2 - 4x + 1?

Solution:

Use the vertex formula x = -b/2a: x = 4/2 = 2. Substitute x back to find y: y = 2^2 - 4(2) + 1 = -3. Vertex is (2, -3).

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Q4. What is the sum of the roots of the quadratic equation 2x^2 + 4x - 6 = 0?

Solution:

Step 1: Use the formula -b/a. Here, b = 4 and a = 2. Step 2: Sum of roots = -4/2 = -2.

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Q5. What is the value of x in the equation 5x^2 - 20 = 0?

Solution:

Add 20 to both sides: 5x^2 = 20. Divide by 5: x^2 = 4. Take the square root: x = ±2.

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Q6. What is the discriminant of the quadratic equation 2x^2 + 4x + 2?

Solution:

Calculate the discriminant: D = b^2 - 4ac = 4^2 - 4(2)(2) = 16 - 16 = 0.