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Factorisation (Quadratic Equation)

Learn different quadratic equation factorization methods like Factors by Grouping, Completing the Square, and using Formulas. Understand how to factorize trinomials and algebraic fractions effectively. Practice solving quadratic equations step by step.

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Factorisation (Quadratic Equation)

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  1. Factorisation (Quadratic Equation)

  2. 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.

  3. 2.7 Factorisation of a Difference of Two Squares Example:

  4. Solution: Do you know? 2.8 Taking out a Common Factor

  5. What is a quadratic trinomial? -It has 3 terms: term, term, and an independent term - a, b and c are constants, and - Eg. 2.9 Factorisation of Quadratic Trinomials -The Distributive Lawis used in reverse to factorise a quadratic trinomial, as illustrated below.

  6. 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.

  7. Note: The product of two linear factors yields a quadratic trinomial; and the factors of a quadratic trinomial are linear factors.

  8. Cross-Multiplication Method

  9. Factorise the following:

  10. Further Quadratic Trinomials Consider

  11. Use of a Common Factor Example: Factorise Take out common factor 2, we have,

  12. 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 Eg. The fraction is improper, if degree of the denominator ≤ degree of the numerator Eg. ,

  13. Example: Simplify (a) (b) (c) Solution: (a) (b) (c)

  14. Addition of Algebraic Fractions Example: Add Solve the following: (a) (b) (c)

  15. 2.11 Solving Quadratic Equations (by Factoring Method) You may solve a quadratic equations using few ways, Factorisation Completed square form Formula of

  16. 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 orx = – 3 (Answer)

  17. 2.12 Solving Quadratic Equation (by Completing the Squares) • Some example of Completed Square form and Perfect Squares: How to express in completed square form?

  18. Eg. Express the following in completed square form

  19. Eg. Express the following in completed square form

  20. . Example: Solve the quadratic equation

  21. . Example: Solve the quadratic equation No solution for real values of x.

  22. . Try the following quadratic equations:

  23. 2.11 Solving Quadratic Equations (by Formulae) Find out: How does this formula formed? From the discriminant , the quadratic equation has One real root/repeated roots if two real roots if (iii) no real roots if

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