Algebra: Solving Quadratic Equations

Definition and Form

• A quadratic equation is a polynomial equation of the second degree.
• It can be written in the standard form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
• The variable 'x' represents an unknown value.

Solving Quadratic Equations

• Factoring: If the quadratic expression can be factored, set each factor equal to zero and solve for x.
• Example: x² + 5x + 6 = 0 factors to (x + 2)(x + 3) = 0, giving solutions x = -2 and x = -3.
• Quadratic Formula: The quadratic formula provides a general solution for any quadratic equation in standard form.
• The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a.

Discriminant

• The discriminant (b² - 4ac) of a quadratic equation determines the nature of its roots.
• Positive Discriminant: Two distinct real roots.
• Zero Discriminant: One real root (a repeated root).
• Negative Discriminant: Two distinct complex roots (non-real roots).

Nature of Roots

• Real and Distinct: The graph of the corresponding quadratic function intersects the x-axis at two distinct points.
• Real and Equal (Repeated Root): The graph of the corresponding quadratic function touches the x-axis at exactly one point.
• Complex (Non-real): The graph of the corresponding quadratic function does not intersect the x-axis.

Completing the Square

• Completing the square is a technique used to rewrite a quadratic equation in a form that facilitates solving.
• It involves manipulating the equation to isolate the squared term and 'complete' the perfect square trinomial.
• This method is sometimes more direct than using the quadratic formula.

Applications

• Quadratic equations are fundamental to modeling various phenomena in physics, engineering, and economics.
• They describe trajectories, projectile motion, areas of shapes, and optimization problems.
• Examples include calculating the height of a thrown object or determining the dimensions of a rectangular area given a constraint.

Relationship Between Coefficients and Roots

• For a quadratic equation ax² + bx + c = 0 with roots r₁ and r₂, the following relationships exist:
  • sum of roots: r₁ + r₂ = -b/a
  • product of roots: r₁ * r₂ = c/a

Graphing Quadratic Equations

• The graph of a quadratic equation is a parabola.
• The parabola opens upward if 'a' is positive, and downward if 'a' is negative.
• The vertex of the parabola is the point where the parabola changes direction.
• The x-coordinate of the vertex is given by x = -b/2a, and the y-coordinate can be calculated by substituting this value of 'x' into the quadratic equation.