Quadratic Equations and Algebraic Manipulation

Quadratic Equations

• A quadratic equation is a polynomial equation of degree two, having the form:
  • ax^2 + bx + c = 0
  • where a, b, and c are constants, and x is the variable
• The quadratic equation has two solutions, also known as roots, which can be real or complex numbers

Algebraic Manipulation

• Factoring:
  • If the quadratic equation can be written in the form (x - r)(x - s) = 0, then the roots are x = r and x = s
  • Factoring is possible when the product of the roots is equal to the constant term (c) and the sum of the roots is equal to the negative of the coefficient of the linear term (b)
• Quadratic Formula:
  • If the equation cannot be factored, the quadratic formula can be used to find the roots:
    • x = (-b ± √(b^2 - 4ac)) / 2a
  • The quadratic formula is derived from the process of completing the square

Formulae Derivation

• Derivation of the Quadratic Formula:
  • Start with the standard form of the quadratic equation: ax^2 + bx + c = 0
  • Divide both sides by a to get: x^2 + (b/a)x + (c/a) = 0
  • Move the constant term to the right-hand side: x^2 + (b/a)x = -c/a
  • Add (b/2a)^2 to both sides to complete the square: x^2 + (b/a)x + (b/2a)^2 = (b/2a)^2 - c/a
  • Simplify the right-hand side: (x + b/2a)^2 = (b^2 - 4ac)/4a^2
  • Take the square root of both sides: x + b/2a = ±√((b^2 - 4ac)/4a^2)
  • Simplify to get the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a

Problem Solving Strategies

• When solving quadratic equations, first try to factor the equation
• If factoring is not possible, use the quadratic formula
• Always check the solutions by plugging them back into the original equation
• Use the discriminant (b^2 - 4ac) to determine the nature of the roots:
  • If the discriminant is positive, the roots are real and distinct
  • If the discriminant is zero, the roots are real and equal
  • If the discriminant is negative, the roots are complex and conjugate

Quadratic Equations

• A quadratic equation is a polynomial equation of degree two in the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable.
• Quadratic equations have two solutions, also known as roots, which can be real or complex numbers.

Algebraic Manipulation

• Factoring: If a quadratic equation can be written in the form (x - r)(x - s) = 0, then the roots are x = r and x = s.
• Factoring is possible when the product of the roots is equal to the constant term (c) and the sum of the roots is equal to the negative of the coefficient of the linear term (b).
• Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a.

Derivation of Quadratic Formula

• The quadratic formula is derived from the process of completing the square.
• Start with the standard form of the quadratic equation: ax^2 + bx + c = 0.
• Divide both sides by a to get: x^2 + (b/a)x + (c/a) = 0.
• Move the constant term to the right-hand side: x^2 + (b/a)x = -c/a.
• Add (b/2a)^2 to both sides to complete the square: x^2 + (b/a)x + (b/2a)^2 = (b/2a)^2 - c/a.
• Simplify to get: x + b/2a = ±√((b^2 - 4ac)/4a^2).
• Simplify to get the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.

Problem Solving Strategies

• Always try to factor the equation when solving quadratic equations.
• If factoring is not possible, use the quadratic formula.
• Check the solutions by plugging them back into the original equation.
• Use the discriminant (b^2 - 4ac) to determine the nature of the roots:
  • Positive discriminant: roots are real and distinct.
  • Zero discriminant: roots are real and equal.
  • Negative discriminant: roots are complex and conjugate.