If the equation x^2 - 5x + k = 0 has equal roots, what is the value of k?

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Question: If the equation x^2 - 5x + k = 0 has equal roots, what is the value of k?

Options:

Correct Answer: 5

Solution:

For equal roots, the discriminant must be zero. Thus, (-5)^2 - 4*1*k = 0 => 25 - 4k = 0 => k = 25/4 = 6.25.

If the equation x^2 - 5x + k = 0 has equal roots, what is the value of k?

If the equation x^2 - 5x + k = 0 has equal roots, what is the value of k?

Step 1: Identify the equation given, which is x^2 - 5x + k = 0.
Step 2: Understand that for the equation to have equal roots, the discriminant must be zero.
Step 3: The discriminant formula for a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac.
Step 4: In our equation, a = 1, b = -5, and c = k.
Step 5: Substitute the values of a, b, and c 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 discriminant equal to zero for equal roots: 25 - 4k = 0.
Step 8: Solve for k by rearranging the equation: 4k = 25.
Step 9: Divide both sides by 4 to find k: k = 25/4.
Step 10: Calculate 25/4, which equals 6.25.

Quadratic Equations – Understanding the properties of quadratic equations, particularly the condition for equal roots which involves the discriminant.
Discriminant – The discriminant of a quadratic equation determines the nature of the roots; for equal roots, it must be zero.