If the roots of the quadratic equation x² - 5x + k = 0 are equal, what is the va

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Question: If the roots of the quadratic equation x² - 5x + k = 0 are equal, what is the value of k? (2023)

Options:

Correct Answer: 6

Exam Year: 2023

Solution:

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

If the roots of the quadratic equation x² - 5x + k = 0 are equal, what is the value of k? (2023)

If the roots of the quadratic equation x² - 5x + k = 0 are equal, what is the value of k? (2023)

Step 1: Identify the quadratic equation, which is x² - 5x + k = 0.
Step 2: Recall that for a quadratic equation ax² + bx + c = 0, the discriminant is given by the formula D = b² - 4ac.
Step 3: In our equation, a = 1, b = -5, and c = k.
Step 4: Substitute the values of a, b, and c into the discriminant formula: D = (-5)² - 4(1)(k).
Step 5: Calculate (-5)², which is 25. So, we have D = 25 - 4k.
Step 6: For the roots to be equal, the discriminant must be zero. Set the discriminant equal to zero: 25 - 4k = 0.
Step 7: Solve for k by rearranging the equation: 4k = 25.
Step 8: Divide both sides by 4 to find k: k = 25 / 4.
Step 9: Calculate 25 / 4, which equals 6.25.

Discriminant of a Quadratic Equation – The discriminant (b² - 4ac) determines the nature of the roots of a quadratic equation. For equal roots, the discriminant must be zero.
Quadratic Formula – The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) is used to find the roots of a quadratic equation.