If the roots of the equation x^2 - 7x + p = 0 are 3 and 4, what is the value of p?

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Q: If the roots of the equation x^2 - 7x + p = 0 are 3 and 4, what is the value of p?

Solution: Using Vieta's formulas, the sum of the roots is 7 and the product is p. Thus, 3 * 4 = p, so p = 12.

Steps: 7

Step 1: Identify the given quadratic equation, which is x^2 - 7x + p = 0.
Step 2: Recognize that the roots of the equation are given as 3 and 4.
Step 3: Use Vieta's formulas, which tell us that the sum of the roots (3 + 4) should equal the coefficient of x (which is -(-7) = 7).
Step 4: Calculate the sum of the roots: 3 + 4 = 7. This confirms the sum is correct.
Step 5: Use Vieta's formulas again, which state that the product of the roots (3 * 4) equals p.
Step 6: Calculate the product of the roots: 3 * 4 = 12.
Step 7: Conclude that p = 12.