If the roots of the equation x^2 - 6x + k = 0 are real and distinct, what is the

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Question: If the roots of the equation x^2 - 6x + k = 0 are real and distinct, what is the range of k? (2020)

Options:

Correct Answer: k < 9

Exam Year: 2020

Solution:

For real and distinct roots, the discriminant must be greater than zero: (-6)^2 - 4*1*k > 0, leading to k < 9.

If the roots of the equation x^2 - 6x + k = 0 are real and distinct, what is the range of k? (2020)

If the roots of the equation x^2 - 6x + k = 0 are real and distinct, what is the range of k? (2020)

Step 1: Identify the quadratic equation given, which is x^2 - 6x + 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 = 1, b = -6, and c = k.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = (-6)^2 - 4*1*k.
Step 5: Calculate (-6)^2, which equals 36. So, D = 36 - 4k.
Step 6: For the roots to be real and distinct, the discriminant must be greater than zero: 36 - 4k > 0.
Step 7: Rearrange the inequality: 36 > 4k.
Step 8: Divide both sides of the inequality by 4: 9 > k.
Step 9: This can also be written as k < 9.
Step 10: Therefore, the range of k for the roots to be real and distinct is k < 9.

Discriminant of a Quadratic Equation – The discriminant (b^2 - 4ac) determines the nature of the roots of a quadratic equation. For real and distinct roots, the discriminant must be greater than zero.
Quadratic Inequalities – Understanding how to manipulate inequalities to find the range of values for parameters in quadratic equations.