Sum and Product of Roots in Quadratics

Roots of Quadratic Equations

• Roots depend on the coefficients of the quadratic equation.
• For the equation (x^2 + 6x - 55 = 0), the coefficients are (a=1), (b=6), and (c=-55).

Sum and Product of Roots

• The sum of the roots for (x^2 + 6x - 55 = 0) is calculated as (-b/a), resulting in (-6).
• The product of the roots for the same equation is (c/a), which gives (-55).

Coefficients in Different Equations

• For the equation (8x^2 - 2x = 1), the coefficients are (a=8), (b=-2), and (c=-1).
• The sum of the roots for (8x^2 - 2x = 1) equals (1/4).
• The product of the roots for this quadratic is (-1/8).

Solution Set

• A potential solution set for (8x^2 - 2x = 1) includes {-1/4, 1/2}.

Additional Quadratic Equations

• For the equation (x^2 + x = 2):
  • The sum of the roots is (-1).
  • The product of the roots is (-2).

Specific Equations with Fractions

• In the quadratic equation (\frac{1}{2}x^2 - \frac{5}{4}x - 3 = 0):
  • The sum of the roots equals (5/2).
  • The product of the roots is (-6).

Roots of Different Variables

• For the equation (5 - 2m - 3m^2 = 0):
  • The sum of the roots is (-2/3).
  • The product of the roots is (-5/3).

Special Cases of Roots

• If the roots of a quadratic equation are given as (1 ± \sqrt{5}), the product of these roots is (-4).