QRT Flashcards
Multiplying Fractions
a/b x x/y
Multiply numerators together and denominator together
= ax/by
Dividing Fractions
a/b ÷ x/y
Flip the fraction you’re dividing by then multiply
= a/b x y/x = ay/bx
Solving a Proportion
x/5 = 3/4
Cross-multiply
4x = 15
4 = 15/4
(xm)(xn)
=xm+n
xm / xn
=xm-n
(xm)n
xmn
(xn)(yn)
xyn
xn / yn
(x/y)n
Multiplying Polynomials
(x2 + 3x + 4)(x + 5)
Multiply each term in the first polynomial with each term in the second
=x2(x + 5) + 3x(x + 5) + 4(x + 5)
=x3 + 5x2 + 3x2 + 15x + 4x + 20
x3 + 8x2 + 19x + 20
Dividing Polynomials
(2x3 + 13x2 + 11x -16) ÷ (x + 5)
Use long division:
= 2x2 + 3x - 4
Factoring
ax + ay
a(x + y)
Factoring
a2 - b2
Difference of squares:
(a + b)(a - b)
eg 4x2 - 9 = (2x + 3)(2x - 3)
Factoring
a2 + 2ab + b2
Squares of binomials:
(a + b)2
Factoring
a2 -2ab + b2
Squares of binomials
(a - b)2
For all x ≠ ±3,
(3x2 - 11x + 6) / (9 - x2)
- Factorising: denominator - difference of squares -
(9 - x2) = (3 - x)(3 + x)
- Factorising: numerator -
(3x2 - 11x + 6) = (x -3)(3x - 2)
- Factorising numerator for -1:
(x -3)(3x - 2) = (-1)(3 - x)(3x - 2)
- Cancelling (3 - x):
(-3x + 2)/(3 + x)
Unknown in a Denominator
1 + 1/x = 2 - 1/x
Cross multiply:
(x + 1)/x = (2x-1)/x
x2 + x = 2x2 - x
2x2 - x2 = x + x
x2 = 2x
Dividing both sides by x:
x = 2
Unknown In an Exponent
If 8x = 16x-1, x = ?
Since both bases are factors of two, can be reexpressed as powers of two:
(23)x = (24)x-1
3x = 4x - 4
x = 4
Quadratic Formula
‘In terms of’
If a = (b + x)(c + x), what is the value of x in terms of a, b and c
b + x = ac + ax
ax - x = b - ac
x(a - 1) = b - ac
x = (b - ac)/(a - 1)
Simulataneous Equations
2x - 9y = 11 and x + 12y = -8
What is the value of x + y?
Add the equations:
3x + 3y = 3
x + y = 1
Absolute Value
|x - 12| = 3
x - 12 = 3
x - 12 = -3
x = 15 or 9
Inequalities
|whatever| < p
-p < |whatever| < +p
Inequalities
|2x - 3| < 7
= -7 < 2x - 3 < 7
2x - 3 > -7, 2x > -4, x > -2
2x - 3 < 7, 2x < 10, x < 5
Sum of Exterior Angles of a Triangle
360º
Triangle Inequality Theorum
The length of any one side of a triangle must be greater than the positive difference between the two other sides and less that their sum
For example, if one side is 3 and another is 7, then the length of the third must be between 4 < x < 10
Similar Triangles
Area od A in tems of B if length of sides of A are twice that of B
A:B = 2:1
Areas = 22:12 = 4:1
Area of an Equilateral Triangle
= (s2√3) / 4
where s is the side length
Special Right Triangles
- 3-4-5
- 5-12-13
- 45º-45º-90º - sides in ratio 1:1:√2 (if sides are 3, 3, x, x = 3√2)
- 30º-60º-90º - sides in ratio 1:√3:2 (if sides 3, 6, x, x = 3√3)
Area of a Trapezoid
= ((base1 + base2) / 2) x height
Area of a Parallelogram
= base x height
Area of a Hexagon
=(3s2√3)/2
Length of an Arc
= (n/360) x 2πr
Area of a Sector
= (n/360) x π r2
Area of a Cone
= π r2 + π rl
r = base radius
l = slant length
Volume of a Cone
= 1/3 π r2 h
Surface Area of a Sphere
= 4π r2
Volume of a Sphere
= 4/3 π r3
Volume of a Pyramid
= 1/3 Bh
B = area of base
h = height
Coordinate Geometry
Distance Between Two Points
√(x1-x2)2 + (y1-y2)2
Slope Intercept
Which of the following has no intercept with the line
y = 4x + 5
(A) y = 0.25x - 5
(B) y = -0.25x -5
(C) y = 4x + 0.2
(D) y = -4x + 0.2
(E) y = -4x -0.5
Must be parallel - gradient 4x
(C)
Cotangent
= 1/tangent
=Adjacent/Opposite
Secant
=1/cos
=Hypotenuse/Adjacent
Cosecant
=1/sine
=Hypotenuse/Opposite
Trigonometry
Important Identities
- sin2 x + cos2 x = 1
- tan2 x + 1 = sec2 x
- 1 + cot2 x = csc2 x
Radian in Degrees
= 180/
Degrees in Radians
One degree = π/180 radians
