Quadratic Equations and Their Solutions

Quadratic Equations

• The general form of a quadratic equation is ( ax^2 + bx + c = 0 ), where ( a \geq 0 ) and the highest power of the variable is 2.

Solving Quadratic Equations

• Factorisation:
  • Involves finding two factors that multiply to give the original quadratic expression.
  • Example: ( x^2 + 5x + 6 = 0 ) can be factored into ( (x+2)(x+3) = 0 ), which gives solutions ( x = -3 ) or ( x = -2 ).
• Formula Method:
  • Uses the quadratic formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
  • The discriminant: ( \Delta = b^2 - 4ac ) determines the nature of the roots.
• Completing Square Method:
  • Involves manipulating the equation to form a perfect square on one side.
  • Steps include:
    • Moving the constant term ( c ) to the right side.
    • Adding and subtracting ( k^2 ) (where ( k = \frac{b}{2} )) to the left side.
    • Taking the square root of both sides and solving for ( x ).

Nature of the Discriminant

• The discriminant, ( \Delta = b^2 - 4ac ), determines the nature of the roots of the quadratic equation:
  • If ( \Delta = 0 ), the roots are real and equal.
  • If ( \Delta > 0 ), the roots are real and unequal.
  • If ( \Delta < 0 ), the roots are not real (they are complex).

Finding a Quadratic Equation

• Given two roots, ( \alpha ) and ( \beta ), the corresponding quadratic equation can be found using the formula:
  • ( x^2 - (\alpha + \beta)x + \alpha\beta = 0 )

Useful Formulas

• Several useful formulas relate to the roots of quadratic equations:
  • ( (\alpha + \beta)^2 = (\alpha^2 + \beta^2) + 2\alpha\beta )
  • ( \alpha^2 + \beta^2 = (\alpha - \beta)^2 + 2\alpha\beta )
  • ( \alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) )
  • ( (\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta )
  • ( a^2 - b^2 = (a + b)(a - b) )