Question

If the sum of the roots of $(p + 1) \times x ^ 2 + (2p + 3)x + (3p + 4) = 0$ is -1 , then the product of the roots is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer:

C. 2

Understanding Quadratic Equations and Their Roots

• A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is squared. Its standard form is ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients and 'a' ≠ 0.

• The solutions to a quadratic equation are called its roots. A quadratic equation always has two roots, which can be real or complex.

Formulas for Sum and Product of Roots

• For a quadratic equation in the form ax² + bx + c = 0:

  • The sum of the roots (α + β) is given by the formula -b/a.

  • The product of the roots (αβ) is given by the formula c/a.

Applying to the Given Problem

• The given quadratic equation is (p + 1)x² + (2p + 3)x + (3p + 4) = 0.

• By comparing this with the standard form ax² + bx + c = 0, we can identify the coefficients:

  • a = (p + 1)

  • b = (2p + 3)

  • c = (3p + 4)

• We are given that the sum of the roots is -1.

• Using the sum of roots formula: -(b/a) = -1.

• Substitute the coefficients: -(2p + 3) / (p + 1) = -1.

• To solve for 'p':

  • (2p + 3) / (p + 1) = 1 (multiplying both sides by -1)

  • 2p + 3 = p + 1 (multiplying both sides by (p + 1))

  • 2p - p = 1 - 3

  • p = -2

• Now that we have the value of p = -2, we can find the product of the roots.

• The product of the roots (αβ) is c/a.

• Substitute the values of 'c', 'a', and 'p':

  • c = 3p + 4 = 3(-2) + 4 = -6 + 4 = -2

  • a = p + 1 = -2 + 1 = -1

• Therefore, the product of the roots = c/a = -2 / -1 = 2.