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Friday, 26 October 2012

ADD MATH FORM 4 :CHAPTER 2









quadratic_equation





Determining Whether a Given Value is a Root of a Quadratic Equation
The roots of a quadratic equation
are the solution for the equation. This means that if the roots are substituted back into the equation, they will satisfy the equation, i.e., the left hand side of the equation (LHS) is equal to the right hand side (RHS) of the equation.
There are two methods to determine whether a given value is a root of a quadratic equation. The methods are:
a) The substitution method
b) The inspection method
The substitution method involves substituting back the given roots into the quadratic equation. If LHS = RHS, the given roots are the root of the quadratic equation.
The inspection method involves factorising the quadratic equation as a product of two linear equations (x + a)(x + b) = 0. By inspection, we can judge whether the given roots make the linear equation zero, thus satisfying the quadratic equation.
Note that since the highest power
of the variable in a quadratic equation is 2, at most there are two roots is a quadratic equation.


topik2 



  • Determining the Roots of a Quadratic Equation
    Determining the roots of a quadratic equation is to find the solution of the quadratic equation.
    The followings are the three ways to determine the roots of a quadratic equation:
    a) Factorisation
    b) Completing the square
    c) Using the quadratic formula

    Factorisation
    The factorisation method involves factorising the quadratic equation as a product of two linear expression to become (ax + b)(cx + d) = 0 . In this case, we have two possibilities as follows:
    formula
    This is because by inspection, we can see that if we put this value back into the quadratic equation, we get 0=0 or LHS = RHS which satisfies the equation. In other words, we get,
    fomula
    root
  • Example Q&A
  • Example: Factorisation Resource

  • Completing the square
    When factorisation does not work in finding the solution, we can use the "completing the square" method. It involves expressing the quadratic equation in the form of (ax+b)2 = c . The roots can thus be obtained by taking the square roots on both sides.



    Using the quadratic formula
    If the factorisation and "completing the square" methods cannot give the roots of a quadratic equation, we need to use the quadratic formula. The quadratic formula is the formula for finding the roots of all quadratic equations. It is given as,
    formula
    where a, b and c are the constants
    of the quadratic equation.


    topik 2. 


    Identifying the Types of Roots of Quadratic Equations
    We already know that the roots of a quadratic equation ax2 + bx + c = 0, where a ≠ 0, can be obtained using the quadratic formula,
    formula
    The term b2- 4ac in the formula is called the discriminant. This discriminant determines the types of roots of quadratic equations as follows:
    a) In b2 -4ac >0 , there are two distinct roots.
    b) In b2 -4ac =0 , there are two equal roots.
    c) In b2 -4ac <0 , there are no roots. 
     
     

    Solving Problems Involving the Use of the Discriminant
    From the discriminant b2 - 4ac , we can get the type of roots of a quadratic equation. From the type of the roots, we can find:
    a) the value of the unknowns in a quadratic equation.
    b) the range of values of the unknowns in a quadratic equation.
    b) the relation between the unknowns in a quadratic equation.















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