The roots of the equation x^2 - 5x + k = 0 are equal. What is the value of k? (2

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Question: The roots of the equation x^2 - 5x + k = 0 are equal. What is the value of k? (2020)

Options:

Correct Answer: 6.25

Exam Year: 2020

Solution:

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

The roots of the equation x^2 - 5x + k = 0 are equal. What is the value of k? (2020)

The roots of the equation x^2 - 5x + k = 0 are equal. What is the value of k? (2020)

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 equal, 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 the roots to be equal: 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: Simplify 25 / 4 to get k = 6.25.

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.