The product of the roots of the equation x^2 - 7x + k = 0 is 10. What is the val

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

Options:

Correct Answer: 17

Solution:

Using Vieta\'s formulas, the product of the roots is k = 10. Thus, k = 17.

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

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

Step 1: Identify the given quadratic equation, which is x^2 - 7x + k = 0.
Step 2: Recall Vieta's formulas, which tell us that for a quadratic equation ax^2 + bx + c = 0, the product of the roots (r1 * r2) is given by c/a.
Step 3: In our equation, a = 1, b = -7, and c = k.
Step 4: According to Vieta's formulas, the product of the roots (r1 * r2) is equal to k/1, which simplifies to k.
Step 5: We know from the problem that the product of the roots is 10.
Step 6: Set k equal to 10, so we have k = 10.
Step 7: However, we need to check if there is a mistake in the short solution provided. The correct interpretation of the product of the roots should lead us to k = 10, not 17.

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 manipulate it to find roots.