Quadratic Equations: Methods and Graphs

Finding Roots of Quadratic Equations

• The quadratic formula, given by (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}), can be used to find the roots of a quadratic equation (ax^2 + bx + c = 0).

Shape of the Graph

• The graph of a quadratic equation is shaped like a parabola, which can open either upwards or downwards depending on the leading coefficient (a).

Coefficients and Real Roots

• The relationship between the coefficients (a), (b), and (c) and the number of real roots is determined by the discriminant (D = b^2 - 4ac):
  • If (D > 0): two distinct real roots.
  • If (D = 0): one real root (a repeated root).
  • If (D < 0): no real roots (complex roots).

Essential Step in Factoring

• An essential step when solving a quadratic equation by factoring is to first express the quadratic in standard form and then find two numbers that multiply to (ac) (the product of coefficients (a) and (c)) and add to (b) (the linear coefficient).

Parabola Opening Upward

• A parabola that opens upward (when (a > 0)) indicates that the quadratic equation has either two real roots if the vertex is above the x-axis, or one real root if the vertex touches the x-axis.