Quadratic Equations: Standard Form and Solutions

• A quadratic equation is a polynomial equation of the second degree.
• The general form is ax² + bx + c = 0, where a ≠ 0.
• x represents a variable or an unknown.
• a, b, and c represent constants, with 'a' not equal to zero; otherwise, the equation would become linear.

Standard Form

• The standard form is ax² + bx + c = 0.
• 'a' is the quadratic coefficient.
• 'b' is the linear coefficient.
• 'c' is the constant term.
• Rearranging the equation into this standard form is the first step in solving for x.

Solutions/Roots

• Solutions to the quadratic equation are also called roots or zeros.
• These values of x satisfy the equation.
• A quadratic equation has exactly two roots, which may be real or complex, and may be equal.

Methods to Solve Quadratic Equations

• Factoring
• Completing the Square
• Quadratic Formula

Factoring

• If the quadratic expression ax² + bx + c can be factored into two linear factors, the solutions can be found by setting each factor to zero.
• Example: x² - 5x + 6 = 0 can be factored into (x - 2)(x - 3) = 0, giving solutions x = 2 and x = 3.
• Factoring is most effective when the roots are rational numbers.

Completing the Square

• Involves transforming the quadratic equation into a perfect square trinomial.
• Useful when the quadratic equation cannot be easily factored.
• Process:
  • Divide the equation by 'a' if a ≠ 1.
  • Move the constant term to the right side of the equation.
  • Add (b/2a)² to both sides of the equation to complete the square.
  • Express the left side as a perfect square and simplify the right side.
  • Take the square root of both sides.
  • Solve for x.
• Example: x² + 6x + 5 = 0
  • x² + 6x = -5
  • x² + 6x + 9 = -5 + 9
  • (x + 3)² = 4
  • x + 3 = ±2
  • x = -3 ± 2, giving solutions x = -1 and x = -5.

Quadratic Formula

• The quadratic formula provides a direct method for finding the solutions of a quadratic equation.
• Given ax² + bx + c = 0, the solutions for x are given by: x = (-b ± √(b² - 4ac)) / (2a).
• The quadratic formula can be used for any quadratic equation, regardless of the nature of its roots.
• Can be derived by completing the square.

Discriminant

• The discriminant is the part of the quadratic formula under the square root: Δ = b² - 4ac.
• It determines the nature of the roots:
  • If Δ > 0, the equation has two distinct real roots.
  • If Δ = 0, the equation has exactly one real root (a repeated root).
  • If Δ < 0, the equation has two complex roots.

Complex Roots

• Complex roots occur when the discriminant is negative.
• They involve the imaginary unit i, where i² = -1.
• Complex roots always occur in conjugate pairs, i.e., if p + qi is a root, then p - qi is also a root.

Sum and Product of Roots

• For a quadratic equation ax² + bx + c = 0, if the roots are x₁ and x₂, then:
  • Sum of roots: x₁ + x₂ = -b/a
  • Product of roots: x₁ * x₂ = c/a
• These relationships can be useful in constructing quadratic equations with specific root properties or verifying solutions.

Applications

• Physics: Projectile motion, trajectory calculations.
• Engineering: Designing parabolic arches, electrical circuits.
• Economics: Modeling cost and revenue functions.
• Computer Graphics: Creating curves and surfaces.

Forming a Quadratic Equation from Roots

• If α and β are the roots of a quadratic equation, the equation can be written as: x² - (α + β)x + αβ = 0.
• This can be derived from the relationships between roots and coefficients.

Graph of a Quadratic Equation

• The graph of a quadratic equation y = ax² + bx + c is a parabola.
• If a > 0, the parabola opens upwards.
• If a < 0, the parabola opens downwards.
• The vertex of the parabola represents the minimum or maximum point of the quadratic function.

Vertex

• The x-coordinate of the vertex is given by x = -b / (2a).
• The y-coordinate of the vertex can be found by substituting this x-value back into the quadratic equation.

Symmetry

• A parabola is symmetric about the vertical line passing through its vertex.
• This line of symmetry is x = -b / (2a).

Intercepts

• The y-intercept is the point where the parabola intersects the y-axis, found by setting x = 0 in the equation (y = c).
• The x-intercepts are the points where the parabola intersects the x-axis, found by solving the quadratic equation ax² + bx + c = 0 (roots of the equation).

Transformations

• The quadratic function can be transformed by changing the values of a, b, and c.
• Vertical stretch/compression: Changing 'a' stretches or compresses the parabola vertically.
• Horizontal shift: Replacing x with (x - h) shifts the parabola horizontally by 'h' units.
• Vertical shift: Adding a constant 'k' to the function shifts the parabola vertically by 'k' units.

Inequalities

• Quadratic inequalities involve comparing a quadratic expression to a value.
• Example: ax² + bx + c > 0 or ax² + bx + c < 0.
• To solve a quadratic inequality:
  • Find the roots of the corresponding quadratic equation.
  • Determine the intervals defined by these roots.
  • Test a value from each interval in the inequality to determine where the inequality holds true.
• Represent the solution set using interval notation.

Systems of Equations

• Quadratic equations can be part of a system of equations, often involving linear equations.
• Solving such systems involves finding the points of intersection between the curves represented by the equations.
• Methods for solving include substitution and elimination.