If the roots of the quadratic equation x^2 - 3x + p = 0 are 1 and 2, what is the

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Question: If the roots of the quadratic equation x^2 - 3x + p = 0 are 1 and 2, what is the value of p?

Options:

Correct Answer: 6

Solution:

Using Vieta\'s formulas, sum of roots = 1 + 2 = 3 and product of roots = 1*2 = 2. Thus, p = 2.

If the roots of the quadratic equation x^2 - 3x + p = 0 are 1 and 2, what is the value of p?

If the roots of the quadratic equation x^2 - 3x + p = 0 are 1 and 2, what is the value of p?

Step 1: Identify the given quadratic equation, which is x^2 - 3x + p = 0.
Step 2: Recognize that the roots of the equation are given as 1 and 2.
Step 3: Use Vieta's formulas, which tell us that the sum of the roots (1 + 2) should equal the coefficient of x (which is -(-3) = 3).
Step 4: Calculate the sum of the roots: 1 + 2 = 3. This matches the coefficient of x, confirming the roots are correct.
Step 5: Now, use Vieta's formulas again to find the product of the roots, which is 1 * 2 = 2.
Step 6: According to Vieta's formulas, the product of the roots also equals p (the constant term in the equation).
Step 7: Therefore, set p equal to the product of the roots: p = 2.

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