The quadratic equation x^2 - 6x + k = 0 has roots that differ by 2. What is the

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Question: The quadratic equation x^2 - 6x + k = 0 has roots that differ by 2. What is the value of k?

Options:

Correct Answer: 10

Solution:

Let the roots be r and r+2. Then, r + (r+2) = 6 and r(r+2) = k. Solving gives k = 10.

The quadratic equation x^2 - 6x + k = 0 has roots that differ by 2. What is the value of k?

The quadratic equation x^2 - 6x + k = 0 has roots that differ by 2. What is the value of k?

Correct Answer: 10

Step 1: Understand that we have a quadratic equation x^2 - 6x + k = 0.
Step 2: Recognize that the roots of the equation differ by 2. Let's call the smaller root 'r' and the larger root 'r + 2'.
Step 3: Use the property of roots that states the sum of the roots (r + (r + 2)) equals the coefficient of x (which is -(-6) = 6). So, we write the equation: r + (r + 2) = 6.
Step 4: Simplify the equation from Step 3: 2r + 2 = 6.
Step 5: Subtract 2 from both sides: 2r = 4.
Step 6: Divide both sides by 2 to find r: r = 2.
Step 7: Now, find the larger root: r + 2 = 2 + 2 = 4.
Step 8: Use the property of roots that states the product of the roots (r * (r + 2)) equals k. So, we calculate: k = r * (r + 2) = 2 * 4.
Step 9: Calculate k: k = 8.
Step 10: Check if the roots differ by 2: 4 - 2 = 2, which is correct.
Step 11: Conclude that the value of k is 8.

Quadratic Equations – Understanding the properties of quadratic equations, including the relationship between roots and coefficients.
Roots of Equations – Using the relationship between the sum and product of the roots to derive equations.
Algebraic Manipulation – Solving for unknowns using algebraic expressions and equations.