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Expanding and factorizing quadratic expressions. Expanding two brackets Squaring expressions The difference between two squares Factorizing expressions Quadratic expressions. Expanding two brackets. Look at this algebraic expression:. (3 + t )(4 – 2 t ).
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Expanding and factorizing quadratic expressions • Expanding two brackets • Squaring expressions • The difference between two squares • Factorizing expressions • Quadratic expressions
Expanding two brackets Look at this algebraic expression: (3 + t)(4 – 2t) This means (3 + t)× (4 – 2t), but we do not usually write × in algebra. To expand or multiply out this expression we multiply every term in the second bracket by every term in the first bracket. (3 + t)(4 – 2t) = 3(4 – 2t) + t(4 – 2t) This is a quadratic expression. = 12 – 6t + 4t – 2t2 = 12 – 2t – 2t2
Expanding two brackets With practice we can expand the product of two linear expressions in fewer steps. For example, – 10 (x – 5)(x + 2) = + 2x – 5x x2 = x2 – 3x – 10 Notice that –3 is the sum of –5 and 2 … … and that –10 is the product of –5 and 2.
Squaring expressions Expand and simplify: (2 – 3a)2 We can write this as, (2 – 3a)2 = (2 – 3a)(2 – 3a) Expanding, 2(2 – 3a) – 3a(2 – 3a) (2 – 3a)(2 – 3a) = = 4 – 6a – 6a + 9a2 = 4 – 12a + 9a2
Squaring expressions In general, (a + b)2 = a2 + 2ab + b2 The first term squared … … plus 2 × the product of the two terms … … plus the second term squared. For example, (3m + 2n)2 = 9m2 + 12mn + 4n2
The difference between two squares Expand and simplify (2a + 7)(2a – 7) Expanding, 2a(2a – 7) + 7(2a – 7) (2a + 7)(2a – 7) = – 49 = – 14a + 14a 4a2 = 4a2 – 49 When we simplify, the two middle terms cancel out. This is the difference between two squares. In general, (a + b)(a – b)= a2 – b2
Writing 5x + 10 as 5(x + 2) is called factorizing the expression. Factorizing expressions Factorize 3x + x2 Factorize 2p + 6p2 – 4p3 The highest common factor of 3x and x2 is The highest common factor of 2p, 6p2 and 4p3 is x. 2p. (2p + 6p2 – 4p3) ÷ 2p = (3x + x2) ÷ x = 3 + x 1 + 3p– 2p2 3x + x2 = x(3 + x) 2p + 6p2 – 4p3 = 2p(1 + 3p– 2p2)
Quadratic expressions t2 ax2 + bx + c (where a = 0) 2 A quadratic expression is an expression in which the highest power of the variable is 2. For example, x2 – 2, w2 + 3w + 1, 4 – 5g2 , The general form of a quadratic expression in x is: x is a variable. a is a fixed number and is the coefficient of x2. b is a fixed number and is the coefficient of x. c is a fixed number and is a constant term.
Expanding or multiplying out a2 + 3a + 2 (a + 1)(a + 2) Factorizing Remember: factorizing an expression is the opposite of expanding it. Factorizing expressions Often: When we expand an expression we remove the brackets. When we factorize an expression we write it with brackets.
Factorizing quadratic expressions Quadratic expressions of the form x2 + bx + c can be factorized if they can be written using brackets as (x + d)(x + e) where d and e are integers. If we expand (x + d)(x + e) we have, (x + d)(x + e) = x2 + dx + ex + de = x2 + (d + e)x + de Comparing this to x2 + bx + cwe can see that: • The sum of d and e must be equal to b, the coefficient of x. • The product of d and e must be equal to c, the constant term.
Factorizing quadratic expressions Quadratic expressions of the form ax2 + bx + c can be factorized if they can be written using brackets as (dx + e)(fx + g) where d, e, f and g are integers. If we expand (dx + e)(fx + g)we have, (dx + e)(fx + g)= dfx2 + dgx + efx + eg = dfx2 + (dg + ef)x + eg Comparing this to ax2 + bx + cwe can see that we must choose d, e, f and g such that: a = df, b = (dg + ef) c = eg
Factorizing the difference between two squares x2 – a2 = (x + a)(x – a) A quadratic expression in the form x2 – a2 is called the difference between two squares. The difference between two squares can be factorized as follows: For example, 9x2 – 16= (3x + 4)(3x – 4) 25a2 – 1= (5a + 1)(5a – 1) m4 – 49n2 = (m2 + 7n)(m2 – 7n)
