Manipulation of Quadratic Expressions

Manipulation of Quadratic Expressions

UNDERSTANDING THE QUADRATIC EXPRESSIONS

  • Quadratic expressions are polynomials of degree 2. They can be in the form ax² + bx + c, where a, b and c are constants and a ≠ 0.
  • The solutions for quadratic expressions are also known as the roots or zeroes of the equations.
  • The term b²-4ac in the quadratic formula is called the discriminant, and it can determine the nature of roots in a quadratic equation.

EXPANDING QUADRATIC EXPRESSIONS

  • When expanding quadratic expressions, care must be taken to use the correct sign.
  • The FOIL method (First, Outside, Inside, Last) can be a useful way to remember how to expand quadratic expressions.
  • For instance, if you have the equation (x + a)(x + b), the expanded form would be x² + (a+b)x + ab.

FACTORISING QUADRATIC EXPRESSIONS

  • Factorising a quadratic expression involves rewriting it in the format (dx + e)(fx + g). This makes it easier to find the roots of the expression.
  • One method to factorise is to look for two numbers that when added together give the coefficient of x, and when multiplied together give the constant term.
  • In general, if you have the expression x² + (a+b)x + ab you can factorise it to (x + a)(x + b).

COMPLETING THE SQUARE

  • Completing the square is another method to represent the quadratic equation in a different form which can help to solve the equation.
  • Given x² + bx + c, completing the square would transform the expression into (x + b/2)² - (b/2)² + c.
  • This method is particularly useful when we need to derive the vertex form of a quadratic equation which gives the minimum or maximum values of an expression.

USING THE QUADRATIC FORMULA

  • The quadratic formula (x = [-b ± sqrt(b² - 4ac)] / 2a) can be used to find the roots of a quadratic expression.
  • The formula is derived from the process of completing the square, and it can help to solve any quadratic equation.
  • Check the value of the discriminant (b² - 4ac): if it’s positive, there are two distinct roots; if it’s zero, there is a repeated root; if it’s negative, there are two complex roots.