stage 8 chapter 18
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Stage 8 Chapter 18. Quadratics. Objectives. Multiply expressions of the form (x+3)(x-7) and simplify the resulting expression; factorise quadratic expressions including the difference of two squares. You should already know. How to collect together simple algebraic terms - PowerPoint PPT PresentationTRANSCRIPT
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Quadratics
Stage 8 Chapter 18
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Objectives
• Multiply expressions of the form (x+3)(x-7) and simplify the resulting expression;
• factorise quadratic expressions including the difference of two squares
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You should already know
• How to collect together simple algebraic terms• Expand single brackets• Take out common factors
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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.
(3 + t)(4 – 2t) = 3(4 – 2t) + t(4 – 2t)
= 12 – 6t + 4t – 2t2
= 12 – 2t – 2t2
This is a quadratic
expression.
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Complete the activityUsing the grid method to expand brackets
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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.
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Complete the activityMatching quadratic expressions 1
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Complete the activityMatching quadratic expressions 2
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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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Complete the activitySquaring 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.
In general,
(a + b)(a – b) = a2 – b2
This is the difference between two squares.
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Complete the activityThe difference between two squares
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Complete the activityMatching the difference between two squares
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Quadratic expressions
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 ,t2
2The 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.
ax2 + bx + c (where a = 0)
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Factorizing expressions
Remember: factorizing an expression is the opposite of expanding it.
Expanding or multiplying out
FactorizingOften:When we expand an expression we remove the brackets.
(a + 1)(a + 2) a2 + 3a + 2
When we factorize an expression we write it with brackets.
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Factorizing quadratic expressions
Factorise x² +7x + 12
This will factorise into two brackets with x as the first term in each
x² +7x + 12 = (x )(x )As both the signs are positive, both the numbers will be positive
You need to find two numbers that multiply together to give 12 and add together to give 7
These will be +3 and +4
So x² +7x + 12 = (x + 3)(x + 4) or x² +7x + 12 = (x + 4)(x + 3)
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Complete the activityFactorizing quadratic expressions 1
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Complete the activityMatching quadratic expressions 1
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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
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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Complete the activityFactorizing quadratic expressions 2
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Matching quadratic expressions 2
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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,
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
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Key idas
• When multiplying two brackets, multiply every term in the first bracket by every term in the second
• To factorise x²+ax+b: if b is positive find two numbers that multiply to give b and add up to a
• To factorise x²+ax+b: if b is negative find two numbers that multiply to give b and have a difference of a
• The difference of two squares factorises x²- a² = (x+a)(x-a)