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.
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.
of the variable in a quadratic equation is 2, at most there are two roots is a quadratic equation.
 Determining the Roots of a Quadratic EquationDetermining 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) Factorisationb) Completing the squarec) Using the quadratic formula
FactorisationThe 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: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,
 Example: Factorisation Resource

Completing the squareWhen 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 formulaIf 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,where a, b and c are the constants
of the quadratic equation.Identifying the Types of Roots of Quadratic EquationsWe already know that the roots of a quadratic equation ax^{2} + bx + c = 0, where a ≠ 0, can be obtained using the quadratic formula,The term b^{2} 4ac in the formula is called the discriminant. This discriminant determines the types of roots of quadratic equations as follows:a) In b^{2} 4ac >0 , there are two distinct roots.b) In b^{2} 4ac =0 , there are two equal roots.c) In b^{2} 4ac <0 , there are no roots.Solving Problems Involving the Use of the DiscriminantFrom the discriminant b^{2}  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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