Summary Quadratics Revision Notes – Cambridge 9709 A Level Mathematics (Worked Examples)

Quadratics
As-A level
mathematics

, Quadratics




I
A quadratic equation is an equation in the form:
Solving quadratic
equations: Example:solving quadratic equations

→ factorisation 2 .
x4+ 7x + 12 = 0 4 (2 -
10x + 21 = 0 7 6x2 + 13x 5 -
= 0


→ complete the (x +4)(x+ 3) = 0 (x-7)(72-3) = 0 (22) -

5)(3x + 1) = 0




square
x -4
=

, x = 3 7) = 7 / 3) = 3 2x -
5 = 0 39 + 1 = 0




5 ,
=
=

→ quadratic formula .
2 (2 - 6x - 16 = 0 .
5 x2- 25 : 0


→ graphically (x 8) (x + 2) 0 x2 52 = 0
- -

=



7) = 8, i =2 (x + 57(x -
5) = 0

x =
- 5 , x= 5
3 . (2+ 52 -
36 = 0
2
(c -
42) + 9x -
36 = 0 .
6 2x2- 11)) + 12 = 0
-




x(x 4) -
+ q(x 4) = 0 -


(2x 3((x - -
4) = 0



(x + 9)(x 4) x= 4
-
-
= 0 x = ,



x
=
-
9 , x = 4




Disguised quadratic equations EXX t4 -
5t2 +6 = 0 EXEx -
45 -

12 = 0


(t2)2 -
572 +6 = 0 let Ja = U
=
4u
(quadratic equation(
> let t = u 12 = 0
Y
- -




↑ incl
(u 6) (U + 2)
-
= 0
2
Let Substitution method
5y + 6 0 V 6 V=
y
-
= = -
z
&

b)
(y 1)(y
= 0
+
-




v2 2 + 1 0 6 5= 6 = 2
y= - y
= -

=




↳
(u -
1)(u -
1) = 0



U = 1


x = 1

x = IT Square root of a
x =
- 1
,
x = 1 number is positive




express Completing the square
: c+ Ex + 11 in the form lital"+6 , always subtract !

S




+
x2+ 6x + 11 s + 6x
L
+ 11 =
(a
(x+ 3)2 +2
(3)2
=

halved +
>
-

> x2+ 6x +9

surplus
>
-
Provided coefficient of x=

I

Ex, j+ 3x +5 =
[x2, 2x2+ 16x -
18 Ex 3/3x -

152 + 9 (xy, -
x 12x
-
+ 20-
(x +
3) + 5
() 2[x + 8x 5] 3[x2 5x + 3) [x + 12x 20]
- -
- - -




2[(x + 4)2 -
5 -
(4)2] 3[(x - E ) + 3 -

(E)Y] -
[(x + 6) -
20 -
16)2]
(x 2)2
+
+ 5 -




q 2[(x+ y)
2
-
5 -
16] 3[(x -

E)2 + 3 -

E] -
[(x + 6)2 -
20 36)
-




(x+ z)z
+
4 7 2[(x+H)2 -
21] ·
Ex-5)2 -]
-
[(x + 6)2 -
56]
>
-
- -

2(x+ 4)2 -
427 3(x 5) -
-

39 -

(x+ 672 + 56
↓ I