Quadratics & Polynomials

Question 1

Complete the Square
example
3x^2 - 12x + 16

Answer 1

remove the common factor
3(x^2 - 4x) + 16
half the value before x
3(x - 2)^2 + 16
take away the 3 times square
3(x - 2)^2 + 16 - 3(4)
3(x - 2)^2 + 4

Question 2

Discriminant - when there is equal roots
2x^2 + kx + 2 = 0

Answer 2

a = 2 b = k c = 2
b^2 - 4ac
(k)^2 - 4(2)(2)
k^2 - 16 = 0
(k + 4)(k - 4)
k = - 4 or k = 4

Question 3

Discriminant - when there is no real roots
x^2 + (k - 6)x + 1 = 0

Answer 3

a = 1 b = k - 6 c = 1
b^2 - 4ac
(k - 6)^2 - 4(1)(1) < 0
k^2 - 12k + 36 - 4 < 0
k^2 - 12k + 32 < 0
(k - 8)(k - 4) < 0
4 < x < 8

Question 4

Discriminant - is a tangent to the curve
y = 2x + 1 is tangent to
y = x^2 + 6x + 5

Answer 4

x^2 + 6x + 5 = 2x + 1
x^2 + 4x + 4

a = 1 b = 4 c = 4
b^2 - 4ac
(4)^2 - 4(1)(4) = 0

Since b^2 - 4ac = 0, the line is a tangent.

Question 5

Solving quadratic inequalities and sketching a graph

example
x^2 - 6x + 5 <0

Answer 5

roots = x^2 - 6x + 5
factorise = (x - 5)(x - 1)
x = 5 x = 1

when is the curve less than zero?
1<x<5

Question 6

Polynomials
example
x^3 + 2x^2 - 5x - 6 (x + 3)

Answer 6

-3 1 2 -5 -6
/ -3 3 6
1 -1 -2 0

Since the remainder = 0, (x + 3) is a factor.

Question 7

Polynomials - factorise fully
example
x^3 - 3x - 2

Answer 7

choose a factor of - 2 (-1, 1, -2,2)
trial and error

2 1 0 -3 -2
/ 2 4 2
1 2 1 0

(x - 2) is a factor
(x - 2) (x^2 + 2x + 1)
(x - 2) (x + 1)(x + 1)

Question 8

Finding the unknown
example
x^3 + 3x^2 + ax - 8 (x - 2)
find the value of a

Answer 8

2 1 3 a -8
/ 2 10 2a +20
1 5 a+10 2a + 12

2a + 12 = 0
2a = -12
a = -6

Question 9

Finding the unknown
example
x^3 + 3x^2 + ax + b (x + 2) (x + 1)
find the value of a and b

Answer 9

-2 1 3 a b -1 1 3 a b
/ -2 a-2 -2a+4 / -1 a -2 a+2
1 1 a-2 b-2a+4 1 2 a-2 b-a+2

b - 2a = -4 b - 2(2) + 4
b - a = -2 b - 4 + 4
-a = -2 b = 0
a = 2

Question 10

Finding the equation from the graph
example
(-2,0) (0,3) (1,0) (3,0)
find the values of k, a, b, c

Answer 10

(-2,0) = (x + 2) (1,0) = (x - 1) (3,0) = (x - 3)
f(x) = k(x + a)(x + b)(x + c)
f(x) = k(x + 2)(x - 1)(x - 3)
3 = k(0 + 2)(0 - 1)(0 - 3)
3 = k x 2 x -1 x -3
6k = 3
k = 1/2