For which value of k does the equation x² - kx + 9 = 0 have no real roots? (2021

For which value of k does the equation x² - kx + 9 = 0 have no real roots? (2021)

For which value of k does the equation x² - kx + 9 = 0 have no real roots? (2021)

Step 1: Identify the equation given, which is x² - kx + 9 = 0.
Step 2: Recognize that to find the roots of a quadratic equation, we use the discriminant formula, which is D = b² - 4ac.
Step 3: In our equation, a = 1, b = -k, and c = 9.
Step 4: Substitute the values into the discriminant formula: D = (-k)² - 4*1*9.
Step 5: Simplify the expression: D = k² - 36.
Step 6: For the equation to have no real roots, the discriminant must be less than zero: k² - 36 < 0.
Step 7: Rearrange the inequality: k² < 36.
Step 8: Take the square root of both sides: |k| < 6.
Step 9: This means k must be between -6 and 6, so k > 6 or k < -6.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots. If the discriminant is less than zero, the equation has no real roots.
Quadratic Equation – A quadratic equation is in the form ax² + bx + c = 0, where a, b, and c are constants.
Inequalities – Understanding how to manipulate inequalities is crucial for determining the range of values for k.