The roots of the equation x² + 2x + k = 0 are real and distinct if k is: (2020)

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Question: The roots of the equation x² + 2x + k = 0 are real and distinct if k is: (2020)

Options:

Correct Answer: < 1

Exam Year: 2020

Solution:

For real and distinct roots, the discriminant must be positive: 2² - 4*1*k > 0, which simplifies to k < 1.

The roots of the equation x² + 2x + k = 0 are real and distinct if k is: (2020)

The roots of the equation x² + 2x + k = 0 are real and distinct if k is: (2020)

Step 1: Identify the equation given, which is x² + 2x + k = 0.
Step 2: Recognize that to find the roots of a quadratic equation, we use the discriminant formula, which is b² - 4ac.
Step 3: In our equation, a = 1, b = 2, and c = k.
Step 4: Substitute the values into the discriminant formula: 2² - 4*1*k.
Step 5: Calculate 2², which equals 4, so we have 4 - 4k.
Step 6: For the roots to be real and distinct, the discriminant must be greater than 0: 4 - 4k > 0.
Step 7: Rearrange the inequality: 4 > 4k.
Step 8: Divide both sides by 4: 1 > k.
Step 9: This means k must be less than 1 for the roots to be real and distinct.

Discriminant of a Quadratic Equation – The discriminant (D) of a quadratic equation ax² + bx + c = 0 is given by D = b² - 4ac. It determines the nature of the roots: if D > 0, the roots are real and distinct; if D = 0, the roots are real and equal; if D < 0, the roots are complex.
Quadratic Inequalities – Understanding how to manipulate inequalities to find the range of values for parameters (like k) that satisfy certain conditions regarding the roots of the quadratic equation.