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C1

1 Simultaneous Equations & Disguised Quadratics

1.1 Linear

1.1.1 Add/ subtract the two equations
to eliminate one variable

1.1.2 Set one equal to y or x and
substitute in

1.2 Quadratic

1.2.1 Substitution

1.3 Disguised Quadratics

1.3.1 Some equations- eg x^4-x^2-5=0 can be converted to quadratic
equations to solve more easily

1.3.2 If there are three orders, ie x^4, x^2, x^0

1.3.3 let A = the middle order, therefore
highest order = A^2

2 Quadratics

2.1 eg 4x^2+3x+7=0

2.2 Solving

2.2.1 If the equation factorises

2.2.1.1 Each bracket = 0

2.2.1.2 eg.(4x+3)(x-2)=0

2.2.1.2.1 x = -3/4 or x = 2

2.2.2 If does not factorise...

2.2.2.1 Quadratic formula

2.2.2.1.1 x= (-b ± √(b^2-4ac) ) / 2a

2.2.2.2 Complete the square

2.2.2.2.1 x^2 ± bx = (x±b/2)^2 - (b/2)^2

2.2.2.2.1.1 Rearrange to find one or two values of x

2.2.2.2.2 a(x+b)^2 + c

2.2.2.2.2.1 Vertex = (-b, c )

2.3 Inequalities

2.3.1 If multiplying or dividing by a negative number, REVERSE the sign

2.3.2 Quadratic

2.3.2.1 Set so that equation = 0

2.3.2.2 Factorise

2.3.2.3 If equation > 0 it is where the graph is above the x axis

2.3.2.4 If equation < 0 it is where the graph is below the x axis

2.4 Intersections of lines

2.4.1 Set equal to eachother to eliminate y

2.4.2 Remember to get the y values at the end by re-substituting the x values

3 Gradients, tangents and normals

3.1 To find a gradient, differentiate the equation and then substitute in the x value

3.2 The tangent to a curve has the same gradient as the point on the curve it touches

3.3 y+y-value= m (x + x-value)

3.4 Stationary points

3.4.1 when dy/dx = 0

3.4.2 solve dy/dx=0 to find stationary points

3.4.3 Differentiate dy/dx to give d^2y/dx^2 .
Substitute in x values, if negative then it
is a max point, if positive it is a min point

4 Coordinate Geometry, Lines and Circles

4.1 Midpoints, gradients and distance between two points

4.1.1 Point A => (x,y) Point B => (w,z)

4.1.1.1 midpoint = ( (x+w)/2 , (y+z)/2 )

4.1.1.2 length of the line through AB = √{ (x+w)^2 + (y+z)^2 }

4.1.1.3 Gradient = (x-w)/(y-z)

4.2 equation of a line through (a,b) with gradient m is y-b = m(x-a)

4.3 Circles

4.3.1 Equation of a circle centre (a,b) radius r = (x-a)^2 + (x-b)^2 = r^2

5 Surds and indices

5.1 Surds

5.1.1 √m x √n = √mn

5.1.2 √m / √n = √(m/n)

5.1.3 To simplify k/√a multiply by √a / √a

5.2 Indices

5.2.1 a^(-n) = 1/(a^n)

5.2.2 a^n x a^m = a^(m+n)

5.2.3 a^m / a^n = a^(m-n)

5.2.4 (a^m)^n = a^(m x n)

5.2.5 a^0 = 1

5.3 a ^ (1/n) = n√a

5.4 a^ (m/n) = n√a^m

6 Curve sketching and transformations

6.1 any graph of the form y=x^n pass through (0,0) and (1,1)

6.2 y=f(x)

6.2.1 y=f(x) + a is a transformation a units upwards

6.2.2 y=f(x+a) is a transformation -a units to the right

6.2.3 y = f(ax) is a stretch sf 1/a parallel to x axis

6.2.4 y = af(x) is a stretch sf a parallel to y axis