Download Quadratic factorisation - Two algorithms

Quadratic factorisation - Two algorithms
Type 1
x2 + bx + c
Example 1
x2 + 5x + 6
Find a pair of numbers with product (+6) and sum (+5)
(+3) and (+2)
x2 + 5x + 6  (x + 3)(x + 2)
Example 2
x2 - x - 2
Find a pair of numbers with product (-2) and sum (-1)
(-2) and (+1)
x2 - x - 2  (x - 2)(x + 1)
Example 3
x2 - 7x + 12
Find a pair of numbers with product (+12) and sum (-7)
(-3) and (-4)
x2 - 7x + 12  (x - 3)(x - 4)
Example 4
x2 - 4
Find a pair of numbers with product (-4) and sum (0)
(-2) and (+2)
x2 - 4 (x - 2)(x + 2)
Try these
1. x2 + 7x + 12
2. x2 + x - 2
3. x2 - 9x + 20
4. x2 - 9
5. x2 - 25
6. x2 - 2x - 8
7. x2 + 7x + 10
8. x2 - 8x + 15
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Quadratic factorisation - Two algorithms
Type 2
ax2 + bx + c
Example 1
6x2 + 5x – 6
• Multiply a by c
6 x (-6) = -36
• Find a pair of numbers with product (-36) and sum b, +5
9 and (-4)
• Write as 9x and -4x
• Replace 5x by 9x and -4x, in either order
• Factorise the red pair and the blue pair
• Factorise by the bracket
6x2 + 9x - 4x – 6
3x(2x + 3) – 2(2x + 3)
6x2 + 5x – 6 = (2x + 3)(3x – 2)
If in the other order at step 4….
6x2 - 4x + 9x – 6
2x(3x - 2) + 3(3x - 2)
6x2 + 5x – 6 = 3x – 2) (2x + 3)
Example 2
12x2 + x – 1
• 12 x (-1) = (-12)
• product (-12) and sum 1
4 and -3
4x and -3x
• 12x + 4x – 3x – 1
2
• 4x(3x + 1) -1(3x + 1)
• 12x2 + x – 1 = (3x + 1)(4x – 1)
Example 3
4x2 + 7x + 3
• 4 x 3 = 12
• product 12 and sum 7
3 and 4
3x and 4x
• 4x + 3x + 4x + 3
2
• x(4x + 3) + 1(4x + 3)
• 4x2 + 7x + 3 = (4x + 3)(x + 1)
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Quadratic factorisation - Two algorithms
Example 4
4x2 – 9
• 4 x (-9) = (-36)
• product (-36) and sum 0
6 and -6
6x and -6x
• 4x2 + 6x – 6x – 9
• 2x(2x + 3) -3(2x + 3)
• 4x2 – 9 = (2x + 3)(2x – 3)
Short cut: a2 - b2 = (a - b)(a + b).
4x2 – 9
a = 2, b = 3
(2x + 3)(2x – 3)
Try these
1. 6x2 + 7x + 2
2. 2x2 + x - 3
3. 3x2 - 5x - 2
4. 9x2 - 4
5. 10x2 + x - 2
6. 3x2 + 10x + 3
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Quadratic factorisation - Two algorithms
Solutions
Type 1
1. x2 + 7x + 12 = (x + 4)(x + 3)
2. x2 + x - 2 = (x - 1)(x + 2)
3. x2 - 9x + 20 = (x - 4)(x - 5)
4. x2 - 9 = (x - 3)(x + 3)
5. x2 - 25 = (x - 5)(x + 5)
6. x2 - 2x - 8 = (x - 4)(x + 2)
7. x2 + 7x + 10 = (x + 2)(x + 5)
8. x2 - 8x + 15 = (x - 3)(x - 5)
Type 2
1. 6x2 + 7x + 2 = (2x + 1)(3x + 2)
2. 2x2 + x - 3 = (x - 1)(2x + 3)
3. 3x2 - 5x - 2 = (3x + 1)(x - 2)
4. 9x2 - 4 = (3x - 2)(3x + 2)
5. 10x2 + x - 2 = (5x - 2)(2x + 1)
6. 3x2 + 10x + 3 = (3x + 1)(x + 3)
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