For the quadratic equation x² + 6x + k = 0 to have no real roots, what must be t

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Question: For the quadratic equation x² + 6x + k = 0 to have no real roots, what must be the value of k? (2021)

Options:

Correct Answer: k > 9

Exam Year: 2021

Solution:

The discriminant must be less than zero: 6² - 4*1*k < 0, which simplifies to k > 9.

For the quadratic equation x² + 6x + k = 0 to have no real roots, what must be the value of k? (2021)

For the quadratic equation x² + 6x + k = 0 to have no real roots, what must be the value of k? (2021)

Step 1: Identify the quadratic equation, which is x² + 6x + k = 0.
Step 2: Recall that for a quadratic equation ax² + bx + c = 0, the discriminant is given by the formula D = b² - 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 = 6² - 4*1*k.
Step 5: Calculate 6², which is 36. So, the discriminant becomes D = 36 - 4k.
Step 6: For the quadratic equation to have no real roots, the discriminant must be less 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 means k must be greater than 9.

Quadratic Equations – Understanding the conditions under which a quadratic equation has real or complex roots, specifically using the discriminant.
Discriminant – The formula used to determine the nature of the roots of a quadratic equation, given by D = b² - 4ac.
Inequalities – Solving inequalities to find the range of values for k that satisfy the condition of having no real roots.