Find the value of k if the equation x^2 + kx + 9 = 0 has no real roots.

Find the value of k if the equation x^2 + kx + 9 = 0 has no real roots.

Step 1: Identify the equation given, which is x^2 + kx + 9 = 0.
Step 2: Understand that for a quadratic equation to have no real roots, the discriminant must be 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 = k, and c = 9. So, the discriminant becomes D = k^2 - 4(1)(9).
Step 5: Simplify the discriminant: D = k^2 - 36.
Step 6: Set the condition for no real roots: k^2 - 36 < 0.
Step 7: Solve the inequality: k^2 < 36.
Step 8: Take the square root of both sides: |k| < 6.
Step 9: This means k must be between -6 and 6: -6 < k < 6.
Step 10: Since we want k to be less than -6 for no real roots, we conclude that k < -6.

Discriminant – The discriminant of a quadratic equation determines the nature of its roots. For the equation ax^2 + bx + c = 0, the discriminant is given by D = b^2 - 4ac. If D < 0, the equation has no real roots.
Quadratic Equations – Quadratic equations are polynomial equations of degree 2, typically in the form ax^2 + bx + c = 0, where a, b, and c are constants.