Mastering Quadratic Equation Factorization Techniques
Learn various factorization techniques for quadratic equations including grouping 'Two and Two', factorization of a difference of two squares, factorization of quadratic trinomials, cross-multiplication method, and use of common factors. Improve your factorization skills and solve quadratic equations efficiently with these methods.
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Factorisation (Quadratic Equation)
2.6 Factors by Grouping 'Two and Two' Now, consider the expression 7x + 14y + bx + 2by. The expression can be grouped into two pairs of two terms as shown. 7(x + 2y) + b(x + 2y) It is evident that (x + 2y) is the common factor. Thus, (x + 2y) + (7 + b) This factorisation technique is called grouping 'Two and Two'; and it is used to factorise an expression consisting of four terms.
Solution: = + + 3 3 2 2 ( )( ) a b a b a ab b Do you know? 2.8 Taking out a Common Factor
2.9 Factorisation of Quadratic Trinomials What is a quadratic trinomial? + x + 2 ax bx c 2 x -It has 3 terms: term, term, and an independent term - a, b and c are constants, and 2 + x x 6 5 + x x - Eg. 0 a 12 + x + + x + 2 2 2 5 6 2 7 3 x x -The Distributive Law is used in reverse to factorise a quadratic trinomial, as illustrated below.
We notice that: 5, the coefficient of x, is the sum of 2 and 3. 6, the independent term, is the product of 2 and 3.
Note: The product of two linear factors yields a quadratic trinomial; and the factors of a quadratic trinomial are linear factors.
Cross-Multiplication Method + x + 2 7 10 x
Further Quadratic Trinomials Consider + x + 2 7 9 2 x
Use of a Common Factor 2 12 14 6 x x Example: Factorise Take out common factor 2, we have, 12 = 2 2 14 6 6 ( 2 7 ) 3 x x x x
2.10 Algebraic Fractions We can write an algebraic fraction in the form Algebraic fraction = Proper and Improper Fraction The fraction is proper, if degree of the denominator > degree of the numerator 3 2+ x x Eg. 4 5 The fraction is improper, if degree of the denominator degree of the numerator x 9 + 2 x 3 1 x Eg. , 2 x 2 + 3
Example: Simplify (a) (b) (c) Solution: (a) (b) (c)
Addition of Algebraic Fractions Example: Add Solve the following: (a) (b) (c)
2.11 Solving Quadratic Equations (by Factoring Method) You may solve a quadratic equations using few ways, (i) Factorisation (ii)Completed square form 2 4 b b 2 ac = x (iii)Formula of a
2.11 Solving Quadratic Equations (by Factoring Method) Eg. Solve x2 + 5x + 6 = 0 x2 + 5x + 6 = 0 (x + 2)(x + 3) =0 x + 2 = 0 or x + 3 = 0 x = 2 or x = 3 (Answer)
2.12 Solving Quadratic Equation (by Completing the Squares) Some example of Completed Square form and Perfect Squares: 2 9 3 9 + = + 2 3 x x x 2 2 4 How to express in completed square form? ( ) 2 = + 2 2 ( ) 2 x a x ax a 2 + = + + 2 2 2 x a x ax a ( ) 2 = 2 2 ( ) 2 a x ax x a a 2 + = + 2 2 2 a x ax x a a = 2 , b = 2 , b 2 2 b b 2 2 b b = 2 x bx x + = + 2 x bx x 2 2 2 2
Eg. Express the following in completed square form + + 2 10 28 x x + + 2 10 28 x x 2 2 10 10 = + + 28 x 2 2 ( ( ) ) 2 = + + 5 25 28 x 2 = + + 5 3 x
Eg. Express the following in completed square form 2 x + 2 10 28 x + 2 2 = 10 28 + x x ( ) 2 2 5 28 x x 2 2 5 5 = + 2 28 x 2 2 2 5 25 = + 2 28 x 2 4 2 5 25 = + 2 28 x 2 2 2 5 31 = + 2 x 2 2
. Example: Solve the quadratic equation
. Example: Solve the quadratic equation ( ) 2 + + = + + 2 10 28 = 5 3 x x x ( ( ) ) 2 + + 5 3 0 x 2 + = 5 3 x No solution for real values of x.
. Try the following quadratic equations:
2.11 Solving Quadratic Equations (by Formulae) 2 4 b b 2 ac = x a Find out: How does this formula formed? 2 From the discriminant , the quadratic equation has ac b 4 ac = 2 4 0 b (i) One real root/repeated roots if ac 2 4 0 b (ii) two real roots if (iii) no real roots if ac 2 4 0 b
