For the quadratic equation x^2 + 4x + k = 0 to have no real roots, k must be:

For the quadratic equation x^2 + 4x + k = 0 to have no real roots, k must be:

Step 1: Identify the quadratic equation, which is x^2 + 4x + 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 = 4, and c = k.
Step 4: Substitute the values into the discriminant formula: D = 4^2 - 4*1*k.
Step 5: Calculate 4^2, which is 16. So, D = 16 - 4k.
Step 6: For the quadratic to have no real roots, the discriminant must be negative: 16 - 4k < 0.
Step 7: Rearrange the inequality: 16 < 4k.
Step 8: Divide both sides by 4 to isolate k: 4 < k.
Step 9: This means k must be greater than 4.

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 = b^2 - 4ac.