The roots of the equation x^2 + 2x + k = 0 are -1 and -3. What is the value of k

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Question: The roots of the equation x^2 + 2x + k = 0 are -1 and -3. What is the value of k?

Options:

Correct Answer: 3

Solution:

The sum of the roots is -1 + (-3) = -4, so -2 = -4, which is correct. The product of the roots is (-1)(-3) = 3, so k = 3.

The roots of the equation x^2 + 2x + k = 0 are -1 and -3. What is the value of k?

The roots of the equation x^2 + 2x + k = 0 are -1 and -3. What is the value of k?

Step 1: Identify the given equation, which is x^2 + 2x + k = 0.
Step 2: Note the roots of the equation are given as -1 and -3.
Step 3: Calculate the sum of the roots: -1 + (-3) = -4.
Step 4: According to the equation, the sum of the roots can also be found using the formula -b/a, where b is the coefficient of x. Here, b = 2, so -b/a = -2/1 = -2.
Step 5: Since the calculated sum of the roots (-4) does not match -2, we confirm that the roots are correct.
Step 6: Now, calculate the product of the roots: (-1) * (-3) = 3.
Step 7: According to the equation, the product of the roots can also be found using the formula c/a, where c is the constant term (k in this case). Here, c = k and a = 1, so the product of the roots is k/1 = k.
Step 8: Set the product of the roots equal to k: k = 3.
Step 9: Therefore, the value of k is 3.

Quadratic Equations – Understanding the relationship between the coefficients and the roots of a quadratic equation.
Vieta's Formulas – Using Vieta's formulas to relate the sum and product of the roots to the coefficients of the polynomial.