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**Expanding and factorizing quadratic expressions**

Expanding two brackets Squaring expressions The difference between two squares Factorizing expressions Quadratic expressions

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**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. In this example, we need to multiply everything in the second bracket by 3 and then everything in the second bracket by t. We can write this as 3(4 – 2t) + t(4 – 2t). As pupils become more confident, they can leave this intermediate step out. (3 + t)(4 – 2t) = 3(4 – 2t) + t(4 – 2t) This is a quadratic expression. = 12 – 6t + 4t – 2t2 = 12 – 2t – 2t2

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**Expanding two brackets**

With practice we can expand the product of two linear expressions in fewer steps. For example, (x – 5)(x + 2) = x2 + 2x – 5x – 10 = x2 – 3x – 10 Notice that –3 is the sum of –5 and 2 … … and that –10 is the product of –5 and 2. Point out that for any expression in the form (x + a)(x + b), where a and b are fixed numbers, the expanded expression will have an x with a coefficient of a + b and the number at the end will be a × b.

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**Matching quadratic expressions 1**

Select a bracketed expression and ask a volunteer to find its corresponding expansion.

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**Matching quadratic expressions 2**

Select a bracketed expression and ask a volunteer to find its corresponding expansion.

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**Squaring expressions Expand and simplify: (2 – 3a)2**

We can write this as, (2 – 3a)2 = (2 – 3a)(2 – 3a) Expanding, (2 – 3a)(2 – 3a) = 2(2 – 3a) – 3a(2 – 3a) = 4 – 6a – 6a + 9a2 = 4 – 12a + 9a2

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**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

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Squaring expressions Any of the terms in the expansion can be hidden or revealed to practice squaring expressions.

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**The difference between two squares**

Expand and simplify (2a + 7)(2a – 7) Expanding, (2a + 7)(2a – 7) = 2a(2a – 7) + 7(2a – 7) = 4a2 – 14a + 14a – 49 = 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

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**Matching the difference between two squares**

Select a bracketed expression and ask a volunteer to find its corresponding expansion.

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**Factorizing expressions**

Writing 5x + 10 as 5(x + 2) is called factorizing the expression. 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)

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**Quadratic expressions**

A quadratic expression is an expression in which the highest power of the variable is 2. For example, t2 2 x2 – 2, w2 + 3w + 1, 4 – 5g2 , The general form of a quadratic expression in x is: ax2 + bx + c (where a = 0) x is a variable. As well as the highest power being two, no power in a quadratic expression can be negative or fractional. Compare each of the quadratic expressions given with the general form. In x2 – 2, a = 1, b = 0 and c = –2. In w2 + 3w + 1, a = 1, b = 3 and c = 1. This is a quadratic in w. In 4 – 5g2, a = –5, b = 0 and c = 4. This is a quadratic in g. In t2/2, a = ½, b = 0 and c = 0. This is a quadratic in t. 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.

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**Factorizing expressions**

Remember: factorizing an expression is the opposite of expanding it. Expanding or multiplying out Factorizing (a + 1)(a + 2) a2 + 3a + 2 Often: When we expand an expression we remove the brackets. When we factorize an expression we write it with brackets.

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**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 Pupils will require lots of practice to factorize quadratics effectively. This slide explains why when we factorize an expression in the form x2 + bx + c to the form (x + d)(x + e) the values of d and e must be chosen so that d + e = b and de = c. (x + d)(x + e) = x2 + (d + e)x + de is an identity. This means that the coefficients and constant on the left-hand side are equal to the coefficients and constant on the right-hand side. Comparing this to x2 + bx + c we 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.

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**Factorizing quadratic expressions 1**

Factorize the given expression by finding two integers that add together to give the coefficient of x and multiply together to give the constant. It may be a good idea to practice adding and multiplying negative numbers before attempting this activity. Use slide 31 in N1.2 Calculating with integers to do this if required. The lower of the two hidden integers will be given first in each case.

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**Matching quadratic expressions 1**

Select a quadratic expression and ask a volunteer to find its corresponding factorization.

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**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 Discuss the factorization of quadratics where the coefficient of x2 is not 1. Most examples at this level will have a as a prime number so that there are only two factors, 1 and the number itself. Comparing this to ax2 + bx + c we can see that we must choose d, e, f and g such that: a = df, b = (dg + ef) c = eg

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**Factorizing quadratic expressions 2**

Factorize each given expression using trial and improvement and the relationships shown on the previous slide. For each expression in the form ax2 + bx + c, start by using the pen tool to write down pairs of integers that multiply together to make a and pairs of integers that multiply together to make c. Use these to complete the factorization.

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**Matching quadratic expressions 2**

Select a quadratic expression and ask a volunteer to find its corresponding factorization.

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**Factorizing the difference between two squares**

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: x2 – a2 = (x + a)(x – a) For example, See slide 37 to demonstrate the expansion of expressions of the form (x + a)(x – a). Pupils should be encouraged to spot the difference between two squares whenever possible. 9x2 – 16 = (3x + 4)(3x – 4) 25a2 – 1 = (5a + 1)(5a – 1) m4 – 49n2 = (m2 + 7n)(m2 – 7n)

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**Factorizing the difference between two squares**

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**Matching the difference between two squares**

Select an expression involving the difference between two squares and ask a volunteer to find the corresponding factorization.

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Expand the brackets. What are the missing coefficients?

Expand the brackets. What are the missing coefficients?

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