For which value of k does the quadratic equation x^2 - kx + 4 = 0 have no real r

For which value of k does the quadratic equation x^2 - kx + 4 = 0 have no real roots?

For which value of k does the quadratic equation x^2 - kx + 4 = 0 have no real roots?

Step 1: Identify the quadratic equation, which is x^2 - kx + 4 = 0.
Step 2: Recall that a quadratic equation has no real roots when its discriminant is 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 = 4. So, the discriminant is D = (-k)^2 - 4(1)(4).
Step 5: Simplify the discriminant: D = k^2 - 16.
Step 6: Set the discriminant less than zero for no real roots: k^2 - 16 < 0.
Step 7: Solve the inequality: k^2 < 16.
Step 8: Take the square root of both sides: |k| < 4.
Step 9: This means k is between -4 and 4: -4 < k < 4.
Step 10: However, we are interested in the values of k that make the discriminant negative, which means k must be greater than 4: k > 4.

Discriminant of a Quadratic Equation – The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac. It determines the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is one real root; and if D < 0, there are no real roots.
Conditions for No Real Roots – For a quadratic equation to have no real roots, the discriminant must be less than zero.