If the roots of the equation x^2 + 5x + k = 0 are real and distinct, what is the

If the roots of the equation x^2 + 5x + k = 0 are real and distinct, what is the condition on k? (2023)

If the roots of the equation x^2 + 5x + k = 0 are real and distinct, what is the condition on k? (2023)

Step 1: Identify the equation given, which is x^2 + 5x + k = 0.
Step 2: Understand that for the roots of a quadratic equation to be real and distinct, the discriminant must be greater than zero.
Step 3: The discriminant (D) for the equation ax^2 + bx + c = 0 is calculated using the formula D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = 5, and c = k. So, we substitute these values into the discriminant formula: D = 5^2 - 4(1)(k).
Step 5: Calculate 5^2, which is 25. Now we have D = 25 - 4k.
Step 6: Set up the inequality for the discriminant to be greater than zero: 25 - 4k > 0.
Step 7: Solve the inequality: First, subtract 25 from both sides: -4k > -25.
Step 8: Divide both sides by -4. Remember, when dividing by a negative number, the inequality sign flips: k < 25.
Step 9: Conclude that for the roots to be real and distinct, the condition on k is k < 25.

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, the roots are real and distinct; if D = 0, the roots are real and equal; if D < 0, the roots are complex.
Conditions for Real and Distinct Roots – For the roots of a quadratic equation to be real and distinct, the discriminant must be positive.