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How to find alpha and beta in a quadratic equation?

Published in Quadratic Equations 2 mins read

Finding alpha (α) and beta (β) in a quadratic equation refers to determining the roots of the equation. These roots are the values of 'x' that satisfy the quadratic equation. Here's how you can find them, drawing on the given information:

Methods for Finding Alpha (α) and Beta (β)

  1. Using the Quadratic Formula:

    The most direct method is to use the quadratic formula, which provides the roots of any quadratic equation in the form ax2 + bx + c = 0. The formula is:

    x = [-b ± √(b2 - 4ac)]/2a

    • One root (α) is found using the plus sign: α = [-b + √(b2 - 4ac)]/2a
    • The other root (β) is found using the minus sign: β = [-b - √(b2 - 4ac)]/2a

  2. Using the Sum and Product of Roots:

    This method is useful if you need to find the relationship between the roots rather than the exact values, or if you know the sum and product of the roots and want to construct the quadratic equation.

    • Sum of Roots: α + β = -b/a
    • Product of Roots: αβ = c/a

    These relationships can be used to solve for α and β if you have additional information or another equation relating them.

  3. Constructing the Quadratic Equation from Roots:

    If you know α and β, you can construct the quadratic equation using the following form:

    x2 - (α + β)x + αβ = 0

    This is the reverse of finding the roots; you're building the equation from the roots themselves.

Examples

Let's say we have the quadratic equation 2x2 + 5x + 3 = 0

  1. Using the Quadratic Formula:

    • a = 2, b = 5, c = 3

    • x = [-5 ± √(52 - 4 2 3)] / (2 * 2)

    • x = [-5 ± √(25 - 24)] / 4

    • x = [-5 ± √1] / 4

    • x = [-5 ± 1] / 4

    • Therefore:

      • α = (-5 + 1) / 4 = -1
      • β = (-5 - 1) / 4 = -3/2

  2. Verifying with Sum and Product of Roots:

    • Sum of roots: α + β = -1 + (-3/2) = -5/2

    • -b/a = -5/2 (Matches the sum of roots)

    • Product of roots: αβ = (-1) * (-3/2) = 3/2

    • c/a = 3/2 (Matches the product of roots)

Table summarizing the formulas

Concept Formula
Quadratic Formula x = [-b ± √(b2 - 4ac)]/2a
Sum of Roots α + β = -b/a
Product of Roots αβ = c/a
Equation from Roots x2 - (α + β)x + αβ = 0

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