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

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

Options:

Correct Answer: 6

Solution:

For the roots to be equal, the discriminant must be zero. Thus, b^2 - 4ac = 0 => 25 - 4k = 0 => k = 25.

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

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

Step 1: Identify the equation given, which is x^2 - 5x + k = 0.
Step 2: Recognize that for the roots of a quadratic equation to be equal, the discriminant must be zero.
Step 3: Write down the formula for the discriminant, which is b^2 - 4ac.
Step 4: Identify the values of a, b, and c from the equation: a = 1, b = -5, c = k.
Step 5: Substitute the values of a and b into the discriminant formula: (-5)^2 - 4(1)(k).
Step 6: Simplify the expression: 25 - 4k.
Step 7: Set the discriminant equal to zero for the roots to be equal: 25 - 4k = 0.
Step 8: Solve for k by adding 4k to both sides: 25 = 4k.
Step 9: Divide both sides by 4 to find k: k = 25 / 4.
Step 10: Conclude that the value of k is 25.

Quadratic Equations – Understanding the properties of quadratic equations, particularly the condition for equal roots.
Discriminant – Using the discriminant (b^2 - 4ac) to determine the nature of the roots of a quadratic equation.