If the roots of the equation x^2 - 6x + k = 0 are 2 and 4, find the value of k.

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Question: If the roots of the equation x^2 - 6x + k = 0 are 2 and 4, find the value of k.

Options:

Correct Answer: 10

Solution:

Using Vieta\'s formulas, k = 2 * 4 = 8.

If the roots of the equation x^2 - 6x + k = 0 are 2 and 4, find the value of k.

If the roots of the equation x^2 - 6x + k = 0 are 2 and 4, find the value of k.

Step 1: Understand that the equation x^2 - 6x + k = 0 is a quadratic equation.
Step 2: Recognize that the roots of the equation are given as 2 and 4.
Step 3: Use Vieta's formulas, which tell us that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots (r1 + r2) is equal to -b/a and the product of the roots (r1 * r2) is equal to c/a.
Step 4: In our equation, a = 1, b = -6, and c = k.
Step 5: Calculate the product of the roots: 2 * 4 = 8.
Step 6: According to Vieta's formulas, the product of the roots (2 * 4) should equal k (since a = 1).
Step 7: Therefore, we find that k = 8.

Vieta's Formulas – Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.
Quadratic Equations – Understanding the standard form of a quadratic equation and how to find its roots.