If the equation x^2 + 5x + k = 0 has no real roots, what must be the condition o

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Question: If the equation x^2 + 5x + k = 0 has no real roots, what must be the condition on k?

Options:

Correct Answer: k < 25

Solution:

The discriminant must be negative for no real roots: 5^2 - 4*1*k < 0, which simplifies to 25 - 4k < 0, or k > 25.

If the equation x^2 + 5x + k = 0 has no real roots, what must be the condition on k?

If the equation x^2 + 5x + k = 0 has no real roots, what must be the condition on k?

Step 1: Identify the equation given, which is x^2 + 5x + k = 0.
Step 2: Understand that for a quadratic equation to have no real roots, the discriminant must be negative.
Step 3: Recall the formula for the discriminant, which is given by D = b^2 - 4ac, where a, b, and c are the coefficients from the equation ax^2 + bx + c.
Step 4: In our equation, a = 1, b = 5, and c = k.
Step 5: Substitute the values into the discriminant formula: D = 5^2 - 4*1*k.
Step 6: Calculate 5^2, which is 25, so we have D = 25 - 4k.
Step 7: Set the condition for no real roots: D < 0, which means 25 - 4k < 0.
Step 8: Rearrange the inequality to find k: 25 < 4k.
Step 9: Divide both sides by 4 to isolate k: k > 25.

Quadratic Equations – Understanding the conditions for real roots based on the discriminant.
Discriminant – The discriminant (b^2 - 4ac) determines the nature of the roots of a quadratic equation.