Algebraic Identities: SSC CGL Quantitative Aptitude

Formulas (2)

• Square of a sum, square of a difference, difference of squares · Square identities
  $$(a + b)^2 = a^2 + 2ab + b^2 \qquad (a - b)^2 = a^2 - 2ab + b^2 \qquad (a + b)^2 + (a - b)^2 = 2(a^2 + b^2) \qquad (a + b)^2 - (a - b)^2 = 4ab \qquad a^2 - b^2 = (a - b)(a + b)$$
• Squares from the sum and product of two numbers · From a sum and a product
  $$a^2 + b^2 = (a + b)^2 - 2ab \qquad (a - b)^2 = (a + b)^2 - 4ab \qquad a^2 - ab + b^2 = (a + b)^2 - 3ab \qquad \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab}$$

Watch out for (5)

• A test value that hides the difference
  → Checking an algebra answer by putting in numbers
• Squaring term by term
  → Square of a sum, square of a difference, difference of squares
• The last term of a squared difference
  → Square of a sum, square of a difference, difference of squares
• Leaving out the coefficient in the product
  → Squares from the sum and product of two numbers
• Two square roots, one answer
  → Squares from the sum and product of two numbers

Formulas (3)

• Cubes from a sum or difference and a product · Cube identities for two terms
  $$(a + b)^3 = a^3 + b^3 + 3ab(a + b) \qquad (a - b)^3 = a^3 - b^3 - 3ab(a - b) \qquad a^3 + b^3 = (a + b)^3 - 3ab(a + b) \qquad a^3 - b^3 = (a - b)^3 + 3ab(a - b)$$
• Spotting an expanded cube · Expanded cubes
  $$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \qquad (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$$
• Sum and difference of cubes as factors · Sum and difference of cubes
  $$a^3 + b^3 = (a + b)(a^2 - ab + b^2) \qquad a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

Watch out for (5)

• The correction carries the sum
  → Cubes from a sum or difference and a product
• The sign in the difference form
  → Cubes from a sum or difference and a product
• Cubing a near-cube
  → Spotting an expanded cube
• Pairing the wrong quadratic
  → Sum and difference of cubes as factors
• Different numbers on top and below
  → Sum and difference of cubes as factors

Formulas (3)

• Squaring x + 1/x and x − 1/x · The squaring ladder
  $$x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2 \qquad x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2 \qquad x^4 + \frac{1}{x^4} = \left(x^2 + \frac{1}{x^2}\right)^2 - 2$$
• Switching between x + 1/x and x − 1/x · Switching the sign
  $$\left(x - \frac{1}{x}\right)^2 = \left(x + \frac{1}{x}\right)^2 - 4 \qquad x^2 - \frac{1}{x^2} = \left(x + \frac{1}{x}\right)\left(x - \frac{1}{x}\right) \qquad x^4 - \frac{1}{x^4} = \left(x^2 + \frac{1}{x^2}\right)\left(x^2 - \frac{1}{x^2}\right)$$
• Getting x + 1/x from an equation or a surd · From a quadratic to x ± 1/x
  $$x^2 - kx + 1 = 0 \Rightarrow x + \frac{1}{x} = k \qquad x^2 - kx - 1 = 0 \Rightarrow x - \frac{1}{x} = k$$

Watch out for (6)

• Plus 2 or minus 2
  → Squaring x + 1/x and x − 1/x
• A number in front left inside
  → Squaring x + 1/x and x − 1/x
• Losing the sign when x is between 0 and 1
  → Switching between x + 1/x and x − 1/x
• Minus 4, not minus 2
  → Switching between x + 1/x and x − 1/x
• Dividing an equation whose ends do not match
  → Getting x + 1/x from an equation or a surd
• A conjugate is the reciprocal only when the product is 1
  → Getting x + 1/x from an equation or a surd

Formulas (3)

• Cubing x + 1/x and x − 1/x · Cubes of x ± 1/x
  $$x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right) \qquad x^3 - \frac{1}{x^3} = \left(x - \frac{1}{x}\right)^3 + 3\left(x - \frac{1}{x}\right) \qquad x^3 + \frac{k^3}{x^3} = \left(x + \frac{k}{x}\right)^3 - 3k\left(x + \frac{k}{x}\right)$$
• Sixth and eighth powers of x ± 1/x · Sixth and eighth powers
  $$x^6 + \frac{1}{x^6} = \left(x^3 + \frac{1}{x^3}\right)^2 - 2 \qquad x^6 + \frac{1}{x^6} = \left(x^2 + \frac{1}{x^2}\right)^3 - 3\left(x^2 + \frac{1}{x^2}\right) \qquad x^8 - \frac{1}{x^8} = \left(x^4 + \frac{1}{x^4}\right)\left(x^2 + \frac{1}{x^2}\right)\left(x^2 - \frac{1}{x^2}\right)$$
• Fifth and seventh powers by multiplying two smaller ones · Product rule for powers of x ± 1/x
  $$\left(x^m + \frac{1}{x^m}\right)\left(x^n + \frac{1}{x^n}\right) = x^{m+n} + \frac{1}{x^{m+n}} + x^{m-n} + \frac{1}{x^{m-n}} \qquad x^5 + \frac{1}{x^5} = \left(x^2 + \frac{1}{x^2}\right)\left(x^3 + \frac{1}{x^3}\right) - \left(x + \frac{1}{x}\right)$$

Watch out for (6)

• Minus 3 times what
  → Cubing x + 1/x and x − 1/x
• The sign in the difference cube
  → Cubing x + 1/x and x − 1/x
• Terms that do not multiply to 1
  → Cubing x + 1/x and x − 1/x
• The wrong correction when cubing a square
  → Sixth and eighth powers of x ± 1/x
• A missing factor in an eighth-power difference
  → Sixth and eighth powers of x ± 1/x
• Taking away the wrong rung
  → Fifth and seventh powers by multiplying two smaller ones

Formulas (3)

• The square of a + b + c · Square of a trinomial
  $$(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$$
• The sum of squared differences · Sum of squared differences
  $$(a - b)^2 + (b - c)^2 + (c - a)^2 = 2(a^2 + b^2 + c^2 - ab - bc - ca)$$
• Squares that add to zero, and least values · Zero sums of squares and least values
  $$a^2 + b^2 + c^2 - ab - bc - ca = 0 \Rightarrow a = b = c \qquad t + \frac{1}{t} \ge 2 \ (t > 0) \qquad (x + y + z)\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) \ge 9$$

Watch out for (4)

• Forgetting to halve
  → The square of a + b + c
• A sum of squared differences below zero
  → The sum of squared differences
• A least value needs the sign
  → Squares that add to zero, and least values
• Equal values need the condition
  → Squares that add to zero, and least values

Formulas (2)

• The factorisation of a³ + b³ + c³ − 3abc · The three-cube identity
  $$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) \qquad a^3 + b^3 + c^3 - 3abc = \frac{1}{2}(a + b + c)\left[(a - b)^2 + (b - c)^2 + (c - a)^2\right] \qquad a^3 + b^3 + c^3 = s^3 - 3sq + 3abc$$
• When a + b + c = 0, a³ + b³ + c³ = 3abc · Three cubes when the sum is zero
  $$a + b + c = 0 \Rightarrow a^3 + b^3 + c^3 = 3abc$$

Watch out for (4)

• Forgetting to add 3abc back
  → The factorisation of a³ + b³ + c³ − 3abc
• The bracket uses the sum of squares
  → The factorisation of a³ + b³ + c³ − 3abc
• Equal values also satisfy the cube condition
  → When a + b + c = 0, a³ + b³ + c³ = 3abc
• A sum that is nearly zero
  → When a + b + c = 0, a³ + b³ + c³ = 3abc

Formulas (2)

• Factorise, then cancel · Factorisations for cancelling
  $$x^2 + (p + q)x + pq = (x + p)(x + q) \qquad a^4 + a^2b^2 + b^4 = (a^2 + ab + b^2)(a^2 - ab + b^2)$$
• Combining fractions and using a given condition · Adding two fractions
  $$\frac{a}{p} + \frac{b}{q} = \frac{aq + bp}{pq}$$

Watch out for (4)

• Cancelling terms, not factors
  → Factorise, then cancel
• A backwards factor changes the sign
  → Factorise, then cancel
• The minus sign before a fraction
  → Combining fractions and using a given condition
• Test values that break the condition
  → Combining fractions and using a given condition