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· 35 min English Formulas At a glance 12 formulas
1 Square of a sum ( a ± b ) 2 = a 2 ± 2 a b + b 2 (a \pm b)^2 = a^2 \pm 2ab + b^2 ( a ± b ) 2 = a 2 ± 2 ab + b 2 a 2 + b 2 = ( a + b ) 2 − 2 a b = ( a − b ) 2 + 2 a b a^2 + b^2 = (a + b)^2 - 2ab = (a - b)^2 + 2ab a 2 + b 2 = ( a + b ) 2 − 2 ab = ( a − b ) 2 + 2 ab 2 Difference of squares a 2 − b 2 = ( a + b ) ( a − b ) a^2 - b^2 = (a + b)(a - b) a 2 − b 2 = ( a + b ) ( a − b ) 3 Cube of a sum ( a + b ) 3 = a 3 + b 3 + 3 a b ( a + b ) (a + b)^3 = a^3 + b^3 + 3ab(a + b) ( a + b ) 3 = a 3 + b 3 + 3 ab ( a + b ) ( a − b ) 3 = a 3 − b 3 − 3 a b ( a − b ) (a - b)^3 = a^3 - b^3 - 3ab(a - b) ( a − b ) 3 = a 3 − b 3 − 3 ab ( a − b ) 4 Factoring cubes a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) a^3 + b^3 = (a + b)(a^2 - ab + b^2) a 3 + b 3 = ( a + b ) ( a 2 − ab + b 2 ) a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 ) a^3 - b^3 = (a - b)(a^2 + ab + b^2) a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) 5 Square of three terms ( a + b + c ) 2 = a 2 + b 2 + c 2 + 2 ( a b + b c + c a ) (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) ( a + b + c ) 2 = a 2 + b 2 + c 2 + 2 ( ab + b c + c a ) 6 Three cubes a 3 + b 3 + c 3 − 3 a b c = ( a + b + c ) S a^3 + b^3 + c^3 - 3abc = (a + b + c)\,S a 3 + b 3 + c 3 − 3 ab c = ( a + b + c ) S S = a 2 + b 2 + c 2 − a b − b c − c a S = a^2 + b^2 + c^2 - ab - bc - ca S = a 2 + b 2 + c 2 − ab − b c − c a 7 Gap form of S S = 1 2 [ ( a − b ) 2 + ( b − c ) 2 + ( c − a ) 2 ] S = \tfrac{1}{2}\left[(a - b)^2 + (b - c)^2 + (c - a)^2\right] S = 2 1 [ ( a − b ) 2 + ( b − c ) 2 + ( c − a ) 2 ] S = 0 exactly when a = b = c S = 0 \text{ exactly when } a = b = c S = 0 exactly when a = b = c 8 Three cubes, zero sum a + b + c = 0 ⇒ a 3 + b 3 + c 3 = 3 a b c a + b + c = 0 \Rightarrow a^3 + b^3 + c^3 = 3abc a + b + c = 0 ⇒ a 3 + b 3 + c 3 = 3 ab c 9 Reciprocal squares x 2 + 1 x 2 = ( x + 1 x ) 2 − 2 x^2 + \dfrac{1}{x^2} = \left(x + \dfrac{1}{x}\right)^2 - 2 x 2 + x 2 1 = ( x + x 1 ) 2 − 2 x 2 + 1 x 2 = ( x − 1 x ) 2 + 2 x^2 + \dfrac{1}{x^2} = \left(x - \dfrac{1}{x}\right)^2 + 2 x 2 + x 2 1 = ( x − x 1 ) 2 + 2 10 Reciprocal cubes x 3 + 1 x 3 = t 3 − 3 t , t = x + 1 x x^3 + \dfrac{1}{x^3} = t^3 - 3t,\ t = x + \dfrac{1}{x} x 3 + x 3 1 = t 3 − 3 t , t = x + x 1 x 3 − 1 x 3 = s 3 + 3 s , s = x − 1 x x^3 - \dfrac{1}{x^3} = s^3 + 3s,\ s = x - \dfrac{1}{x} x 3 − x 3 1 = s 3 + 3 s , s = x − x 1 11 Squares adding to zero Complete each square, e.g. a 2 − 2 a + 1 = ( a − 1 ) 2 a^2 - 2a + 1 = (a - 1)^2 a 2 − 2 a + 1 = ( a − 1 ) 2 12 Make a square Add and subtract a term to build a square, then split with a 2 − b 2 a^2 - b^2 a 2 − b 2 x 4 + 4 = ( x 2 + 2 ) 2 − ( 2 x ) 2 x^4 + 4 = (x^2 + 2)^2 - (2x)^2 x 4 + 4 = ( x 2 + 2 ) 2 − ( 2 x ) 2 Test yourself Q1 Easy
If x − 1 x = 4 x - \dfrac{1}{x} = 4 x − x 1 = 4 x 2 + 1 x 2 x^2 + \dfrac{1}{x^2} x 2 + x 2 1
A 14 B 16 C 18 D 20
· 18
Why
Square the given difference: the middle term is 2 × x × 1 x = 2 2 \times x \times \frac{1}{x} = 2 2 × x × x 1 = 2
( x − 1 x ) 2 = x 2 + 1 x 2 − 2 = 16 \left(x - \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} - 2 = 16 ( x − x 1 ) 2 = x 2 + x 2 1 − 2 = 16
So x 2 + 1 x 2 = 18 x^2 + \frac{1}{x^2} = 18 x 2 + x 2 1 = 18
Q2 Easy
What is the value of ( 6.37 ) 2 − ( 3.63 ) 2 2.74 \dfrac{(6.37)^2 - (3.63)^2}{2.74} 2.74 ( 6.37 ) 2 − ( 3.63 ) 2
A 1 B 2.74 C 6.37 D 10
· 10
Why
Spot the difference of squares on top.
( 6.37 ) 2 − ( 3.63 ) 2 = ( 6.37 + 3.63 ) ( 6.37 − 3.63 ) = 10 × 2.74 (6.37)^2 - (3.63)^2 = (6.37 + 3.63)(6.37 - 3.63) = 10 \times 2.74 ( 6.37 ) 2 − ( 3.63 ) 2 = ( 6.37 + 3.63 ) ( 6.37 − 3.63 ) = 10 × 2.74
Dividing by 2.74 leaves 10.
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