The sum of the roots of the equation x^2 - 7x + k = 0 is 7. What is the value of

₹0.0

Question: The sum of the roots of the equation x^2 - 7x + k = 0 is 7. What is the value of k if the product of the roots is 10? (2023)

Options:

Correct Answer: 10

Exam Year: 2023

Solution:

Using Vieta\'s formulas, the sum of the roots is 7 and the product is k = 10.

The sum of the roots of the equation x^2 - 7x + k = 0 is 7. What is the value of k if the product of the roots is 10? (2023)

The sum of the roots of the equation x^2 - 7x + k = 0 is 7. What is the value of k if the product of the roots is 10? (2023)

Step 1: Identify the equation given, which is x^2 - 7x + k = 0.
Step 2: Recall Vieta's formulas, which state that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots (r1 + r2) is -b/a and the product of the roots (r1 * r2) is c/a.
Step 3: In our equation, a = 1, b = -7, and c = k.
Step 4: According to Vieta's formulas, the sum of the roots (r1 + r2) = -(-7)/1 = 7. This matches the information given in the question.
Step 5: Next, we know the product of the roots (r1 * r2) = k. The question states that this product is equal to 10.
Step 6: Therefore, we can set k = 10.
Step 7: Conclude that the value of k is 10.

Vieta's Formulas – These 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 manipulate it to find roots.