Y12 Algebra & Proof

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  • Created by: fs20/22
  • Created on: 28-10-20 09:01

Quadratics

Solving Quadratics: 

1. Quadratic formula Ink Drawings
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2. Completing the square 

3. Factorising 

4. Sketch

5. Calculator (menu - A:Equation - 2:Polynomial)

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Quadratics EXAMPLES

Factorising 

1. x+ 5x + 6 = 0          (x + 3)( x + 2) = 0          x = -3 or x = -2

2. x- 36 = 0  difference of two squares          (x + 6)(x - 6) = 0          x = -6 or x = 6

Completing the Sqare      y = (x + 1/2 b)- 1/2 b+ c       

Examples:

1. y = x+ 4x + 1        y = (x + 2)- 2+ 1        y = (x + 2)-3        Minimum point = (-2, -3)

2. y = 5x+ 20x - 8          y = 5[x+ 4] - 8          y = 5[(x + 2)-22 ] -8          y = 5(x + 2)- 28

Minimum point = (-2, -28)

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Discriminant

The part of the quadratic formula that determines the the number of solutions to an equation is the part under the square root sign: b- 4ac (the discriminant).

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  • When b- 4ac > 0, there are two real & distinct solutions (crosses x-axis at two points).
  • When b- 4ac = 0, there is one, repeated solution (just touches x-axis at one point). 
  • When b- 4ac < 0, there are no real solutions (doesn't cross x-axis). 

Examples:       1. Find the discriminant & solve the equation 4x2 + 12x + 9 = 0

discriminant = b- 4ac = 12- 4 x 4 x 9 = 0    b- 4ac = 0, hence one solution     x = -3/2 

2. Find the value of the constant p such that he equation x+ x + p has equal roots. 

a = 1, b = 1, c = p        b- 4ac = 0        1- 4 x 1 x p = 0        1 - 4p = 0        1 = 4p        p = 1/4

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Inequalities

Solving Inequalities: 

1. Rearranging equation            e.g. 3x + 1 > x - 5           2x > -6          x > -3

NB: if you multiply or divide by a negative number, flip the sign

2. Factoristing & drawing graph    e.g. x- 36x ≤ 0          (x + 6)(x - 6) ≤ 0        x = 6 or -6

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            -6 ≤ x ≤ 6 

3. Calculator  (menu - B: Inequality)

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Indices

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Example:

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Surds

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Rationalising the Denominator

Example: 

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Simultaneous Equations

Solving simultaneous equations: 

1. Elimination

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2. Substitution 

e.g. Solve y = x and y = 3x- 2

x = 3x- 2         0 = 3x- x - 2         0 = (3x + 2)(x - 1)         x = -3/2, y = -3/2 or x = 1, y = 1

3. Calculator (menu - A:Equation - 1:Simul Equation) 

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