The quadratic equation x^2 + 6x + k = 0 has no real roots. What is the condition

₹0.0

Question: The quadratic equation x^2 + 6x + k = 0 has no real roots. What is the condition on k? (2020)

Options:

Correct Answer: k > 9

Exam Year: 2020

Solution:

For no real roots, the discriminant must be less than zero: 6^2 - 4*1*k < 0, which gives k > 9.

The quadratic equation x^2 + 6x + k = 0 has no real roots. What is the condition on k? (2020)

The quadratic equation x^2 + 6x + k = 0 has no real roots. What is the condition on k? (2020)

Step 1: Identify the quadratic equation, which is x^2 + 6x + k = 0.
Step 2: Recall that a quadratic equation has no real roots if its discriminant is less than zero.
Step 3: The discriminant (D) for the equation ax^2 + bx + c = 0 is given by the formula D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = 6, and c = k.
Step 5: Substitute the values into the discriminant formula: D = 6^2 - 4*1*k.
Step 6: Simplify the expression: D = 36 - 4k.
Step 7: Set the condition for no real roots: 36 - 4k < 0.
Step 8: Solve the inequality: 36 < 4k.
Step 9: Divide both sides by 4: 9 < k.
Step 10: Rewrite the condition: k > 9.

Discriminant of a Quadratic Equation – The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac. It determines the nature of the roots: if D < 0, there are no real roots.
Conditions for Real Roots – Understanding the conditions under which a quadratic equation has real roots is crucial, specifically that the discriminant must be non-negative for real roots to exist.