If one root of the equation x^2 - 7x + k = 0 is 3, what is the value of k?

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Question: If one root of the equation x^2 - 7x + k = 0 is 3, what is the value of k?

Options:

Correct Answer: 9

Solution:

Using Vieta\'s formulas, if one root is 3, the other root is 7 - 3 = 4. Thus, k = 3 * 4 = 12.

If one root of the equation x^2 - 7x + k = 0 is 3, what is the value of k?

If one root of the equation x^2 - 7x + k = 0 is 3, what is the value of k?

Step 1: Start with the quadratic equation x^2 - 7x + k = 0.
Step 2: We know that one root of the equation is 3.
Step 3: According to Vieta's formulas, the sum of the roots of the equation is equal to the coefficient of x (which is -(-7) = 7).
Step 4: If one root is 3, we can find the other root by subtracting 3 from the sum of the roots: 7 - 3 = 4.
Step 5: Now we have both roots: 3 and 4.
Step 6: The product of the roots (which is equal to k) is found by multiplying the two roots together: 3 * 4 = 12.
Step 7: Therefore, the value of k is 12.

Quadratic Equations – Understanding the properties of quadratic equations and how to find roots using Vieta's formulas.
Vieta's Formulas – Using Vieta's relations to relate the coefficients of a polynomial to sums and products of its roots.