Algebra
Solve for x:
x2 - 7x + 12 = 0
x = 3, 4
Simplify 2 root 8 x root 12
2 x 2 root 2 x 2 root 3
= 8 root 6
Sketch the graph of y = x2 - 7x
Show all important features such as intercepts, vertex and axis of symmetry.
AOS: x = 3.5
Vertex: (3.5, -12.25)
x-intercepts: x = 0, 7
Given f(x) = x2 + 1, find f(a - 1).
f(a - 1) = (a - 1)2 + 1 = a2 - 2a + 1 + 1
= a2 - 2a + 2
A ladder 10m long leans against a vertical wall and makes an angle of 60˚ with the ground. How high up does the ladder reach? Leave your answer as exact.
5 root 3 metres
Simplify (9x2 - y2)/(6xy - 2y2)
(3x + y)(3x - y) / 2y(3x - y)
= 3x + y / 2y
root 8 - root 18 + root 50
2 root 2 - 3 root 2 + 5 root 2
= 4 root 2
State the vertex of:
y = 2x2 + 4x - 10
x = -4/4 = -1
y = 2(-1)2 + 4(-1) - 10 = 2 - 4 - 10 = -12
Vertex: (-1, -12)
State the natural domain of root (1 - x)
(-infinity, 1]
If tanx = 4/3, find the exact value of cosx
cosx = 3/5
Factorise 6y3 + 26y2 + 8y
2y(3y2 + 13y + 4)
= 2y(3y + 1)(y + 4)
Rationalise 6/root 3
6 root 3/3 = 2 root 3
Find the exact solutions to 2x2 - 6x + 3 = 0
x = 6 +/- root (36 - 4(2)(3)) / 4
x = 6 +/- root 12 / 4
x = 6 +/- 2 root 3 / 4
x = 3 +/- root 3 / 2
Determine whether f(x) = 2x2 - 5 is odd, even or neither.
f(-x) = 2(-x)2 - 5 = 2x2 - 5
-f(x) = -(2x2 - 5) = -2x2 + 5
f(-x) ≠ -f(x) & f(x) ≠ f(-x)
Therefore f(x) is neither
Madie is standing 18m from the base of a building. Her angle of elevation to the top of the building is 30˚. Calculate the height of the building. Leave your answer as exact.
height = 6 root 3 metres
Simplify
(x2 - 5x + 6)/(x2 - 9) x (x2 + 3x)(x2 - x - 2)
(x-3)(x-2)/(x+3)(x-3) x x(x+3)/(x-2)(x+1)
= x/(x + 1)
Given (6 - root 2)2 = a - b root 2, find the values of a and b
(6 - root 2)2 = 36 - 12 root 2 + 2 = 38 - 12 root 2
a = 38, b = 12
Sketch the graph of y = x(3 - x)(x + 1)
Negative cubic
x-intercepts: x = -1, 0, 3
Given f(x) = 2x + 3 and g(x) = 5x + b,
find the value of b such that f(g(x)) = g(f(x))
f(g(x)) = 2(5x + b) + 3 = 10x + 2b + 3
g(f(x)) = 5(2x + 3) + b = 10x + 15 + b
10x + 2b + 3 = 10x + 15 + b
b = 12
Calculate the area of a triangle with side lengths 6cm, 7cm and 11cm (2 decimal places).
Find angle first (115˚23')
A = 1/2 x sin115˚23' x 6 x 7
= 18.97 cm2
Express as a single fraction:
1/(x2 - 5x + 6) - 1/(x2 + 2x - 8)
1/(x-3)(x-2) - 1/(x+4)(x-2)
= x+4 - (x-3) / (x-3)(x-2)(x+4)
= 7 / (x-3)(x-2)(x+4)
Simplify 128x / 163x + 1
27 / 24(3x + 1)
= 27 / 212x + 4
= 23-12x
State the equation of the line that passes through (4, -7) and makes an angle of inclination of 135˚ with the x-axis. Give your answer in general form.
m = -1
y - -7 = -1(x - 4)
y + 7 = -x + 4
x + y + 3 = 0
State the centre and radius of:
3x2 - 6x + 3y2 + 9y = 81
(x - 1)2 + (y + 3/2)2 = 27 + 1 + 9/4
Centre: (1, -3/2)
Radius: 11/2 = 5.5
Maggie is standing on a 5 metre tall building. Her angle of elevation to the top of a taller building opposite her is 32˚ and the angle of depression to the bottom of the building is 21˚. Calculate the distance between the two buildings (2 decimal places).
tan 21 = 5/x
x = 13.04 metres