Cambridge International Mathematics

EXERCISE 21B.1

1 Solve for the unknown using the Null Factor law:

a 3 x=0 b a£8=0 c ¡ 7 y=0

d ab=0 e 2 xy=0 f abc=0

g a^2 =0 h pqrs=0 i a^2 b=0

2 Solve forxusing the Null Factor law:

a x(x¡5) = 0 b 2 x(x+3)=0 c (x+ 1)(x¡3) = 0

d 3 x(7¡x)=0 e ¡ 2 x(x+1)=0 f 4(x+ 6)(2x¡3) = 0

g (2x+ 1)(2x¡1) = 0 h 11(x+ 2)(x¡7) = 0 i ¡6(x¡5)(3x+2)=0

j x^2 =0 k 4(5¡x)^2 =0 l ¡3(3x¡1)^2 =0

STEPS FOR SOLVING QUADRATIC EQUATIONS

To use theNull Factorlaw when solving equations, we must have one side of the equationequal to zero.

Step 1: If necessary, rearrange the equation so one side iszero.

Step 2: Fully factorisethe other side (usually the LHS).

Step 3: Use theNull Factorlaw.

Step 4: Solvethe resulting linear equations.

Step 5: Checkat least one of your solutions.

Example 4 Self Tutor

If a£b=0

then either

a=0or

b=0.

Solve forx: x^2 =3x

x^2 =3x

) x^2 ¡ 3 x=0 frearranging so RHS=0g

) x(x¡3) = 0 ffactorising the LHSg

) x=0or x¡3=0 fNull Factor lawg

) x=0or x=3

) x=0or 3

ILLEGAL CANCELLING

Let us reconsider the equation x^2 =3x fromExample 4.

We notice that there is a common factor ofxon both sides.

If we cancelxfrom both sides, we will have

x^2

x

=

3 x

x

and thus x=3:

Consequently, we will ‘lose’ the solution x=0.

From this example we conclude that:

We must never cancel a variable that is a common factor from both sides of an

equation unless we know that the factor cannot be zero.

424 Quadratic equations and functions (Chapter 21)

IGCSE01

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Y:\HAESE\IGCSE01\IG01_21\424IGCSE01_21.CDR Monday, 27 October 2008 2:08:58 PM PETER