Math

• Quadratic equation


  Second-degree algebraic expression in the form ax^2 + bx + c = 0, where a, b, and c are coefficients and x is the variable
• Quadratic equation


  • The coefficient of x^2 is a non-zero term (a ≠ 0)
  • Can be presented in different forms but needs to be transformed into standard form before solving
• Roots of a quadratic equation

  The two values of x that satisfy the equation
• Finding roots of a quadratic equation


  1. Using the quadratic formula

  2. Factorizing the equation

  3. Completing the square

  4. Graphing the equation
• Quadratic formula


  x = (-b ± √(b^2 - 4ac)) / 2a
• Nature of roots


  • If discriminant D = b^2 - 4ac > 0, roots are real and distinct
  • If D = 0, roots are real and equal
  • If D < 0, roots are imaginary
• Sum of roots
• Product of roots
• Quadratic equation from given roots

  x^2 - (α + β)x + αβ = 0
• Formulas related to quadratic equations

  • Quadratic equation in standard form: ax^2 + bx + c = 0
  • Discriminant: D = b^2 - 4ac
  • Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
  • Sum of roots: α + β = -b/a
  • Product of roots: αβ = c/a
  • Equation from given roots: x^2 - (α + β)x + αβ = 0
  • Condition for same roots: (a1b2 - a2b1)(b1c2 - b2c1) = (a2c1 - a1c2)^2
  • Minimum/maximum value: x = -b/2a
• Domain of quadratic function
• Graphing Method to Find the Roots


  Look in detail at each of the above methods to understand how to use these methods, their applications, and their uses
• Solving Quadratic Equations by Factorization

  1. Factorization of quadratic equation follows a sequence of steps

  2. Split the middle term into two terms, such that the product of the terms is equal to the constant term

  3. Take the common terms from the available term, to finally obtain the required factors
• Method of Completing the Square

  1. Algebraically square and simplify, to obtain the required roots of the equation

  2. Express the left-hand side as a perfect square, by introducing a new term (b/2a)^2 on both sides

  3. Use the quadratic formula to obtain the roots
• Graphing a Quadratic Equation


  1. Represent the quadratic equation as a function y = ax^2 + bx + c

  2. Solve and substitute values for x to obtain values of y

  3. Present the points in the coordinate axis to obtain a parabola-shaped graph

  4. The point(s) where the graph cuts the horizontal x-axis is the solution of the quadratic equation
• Quadratic Equations Having Common Roots
• Condition for two quadratic equations having common roots

  (a1b2 - a2b1) (b1c2 - b2c1) = (a2c1 - a1c2)^2
• Maximum and Minimum Value of Quadratic Expression

  For a > 0, the quadratic expression has a minimum value at x = -b/2a

  For a < 0, the quadratic expression has a maximum value at x = -b/2a

  x = -b/2a is the x-coordinate of the vertex of the parabola

  For a > 0, Range: [f(-b/2a), ∞)

  For a < 0, Range: (-∞, f(-b/2a)]
• Tips and Tricks on Quadratic Equation
• The roots of a quadratic equation are also called the zeroes of the equation
• For quadratic equations having negative discriminant values, the roots are represented by complex numbers
• The sum and product of the roots of a quadratic equation can be used to find higher algebraic expressions involving these roots
• Definition of a Quadratic Equation

  A quadratic equation in math is a second-degree equation of the form ax^2 + bx + c = 0. Here a and b are the coefficients, c is the constant term, and x is the variable.
• Quadratic Formula


  x = [-b ± √(b^2 - 4ac)]/2a
• How to Solve a Quadratic Equation

  Solve by factorization

  Use the quadratic formula when factorization is not possible
• Quadratic equation


  A second-degree equation of the form ax^2 + bx + c = 0, where a, b, and c are coefficients and x is the variable
• Quadratic equation


  • Has two roots or answers
• Solving a quadratic equation

  1. Factorizing

  2. Using the quadratic formula

  3. Completing the square
• Discriminant


  The value b^2 - 4ac, which determines the nature of the roots of the quadratic equation
• Discriminant value


  D > 0: Real and distinct roots

  D = 0: Real and equal roots

  D < 0: Imaginary/complex roots
• Real-life applications of quadratic equations

  • Running time problems

  Demand and cost calculations

  Satellite dish or reflecting telescope shape
• Linear equation


  An equation of the form ax + b = 0, with one degree and one variable
• 4 ways to solve a quadratic equation

  • Factorizing

  Using the quadratic formula

  Completing the squares

  Graphing
• Solving a quadratic equation by completing the square


  Use the formula (a + b)^2 = a^2 + 2ab + b^2
• The discriminant b^2 - 4ac helps predict the nature of the roots without actually finding them
• Quadratic equations can be solved by graphing, where the x-intercepts are the solutions
• There are three methods to find the roots of a quadratic equation: factorization, completing the square, and using the quadratic formula
• Deriving the quadratic formula

  Use the algebraic formula (a + b)^2 = a^2 + 2ab + b^2
• The quadratic formula is derived using the algebraic formula (a + b)^2 = a^2 + 2ab + b^2