1. Quadratics
ax2 + bx + c

2. Multiplying brackets (FOIL)x2
+2x
+3x +6
x2 +5x +6
Outside
( x+ 3)(x + 2)
Inside
Last
First

3. Multiplying brackets (FOIL) (with minus numbers)x2
-2x
+4x -8
x2 +2x -8
Outside
( x+ 4)(x - 2)
Inside
Last
First

4. Multiplying brackets (FOIL) (with minus numbers)x2
-4x
-3x
+12
x2 -7x +12
Outside
( x-3)(x - 4)
Inside
Last
First

5. y
y = x2 +3x -4
When y =0
X = 1 or -4 x

6. You can find a quadratic from it’s roots
Quadratics
You can find a quadratic from it’s roots

7. Quadratics
For example, x = 3 or -2
When y =0 (x - 3) =0
or (x + 2 ) =0
Multiply the brackets
x2 - x - 6 = 0

8. Factorising and solving quadratics
The x’s have to multiply to make the first term
x2 +5x +6
( x )(x ) +3 +2
The numbers have to add up to +5
and multiply to make +6

9. Check it out Outside ( x+ 3)(x + 2) Last Inside First
x2 +2x + 3x = (x2 + 5x +6)

10. Factorising and solving quadratics
The x’s have to multiply to make the first term
2x2 - 4x - 6
2x x -3 = -6x
-6x +2x = -4x
( 2x )(x ) +2
- 3
The numbers have to add up to -4
and multiply to make -6

11. Examples to remember
(a – b)(a + b)
a2 +ab –ab – b2
= a2 – b2

12. The same applies to all these type of equations
Examples to remember
The same applies to all these type of equations
a2 – 9
= (a-3)(a+3)

13. The same applies to all these type of equations
Examples to remember
The same applies to all these type of equations
4a2 – 36
= (2a -6)(2a+6)

14. Solving using the Quadratic formula
ax2 +bx +c
is the standard form of a quadratic equation (where a, b and c represent numbers)
to find x use the equation
x = (-b ± √(b2 – 4ac))/2a

15. Solving using the Quadratic formula
2x2 +8x + 6 = 0
a = 2, b = 8 , c = 6
x = (-8 ± √(82 – 4*2*6))/2*2
=(-8 ± √(64 – 48))/4
= ( -8 ± √16)/4
(-8 ± 4)/4
= -12/4 or -4/4
= -3 or -1

16. Solving using the Quadratic formula
x2 -10x + 34 = 0
x = (10 ± √(102 – 4*1*-34))/2
=(10 ± √(100 – 136))/2
= ( 10 ± √-36)/2
(10 ± 6i)/2
= 5 + 3i or 5 – 3i
i is an imaginary number (i2 = -1)
Cannot have
a zero square
number so has
to be multiplied
by i2 to make it
positive