Algebra formulas revision notes

The cards

1. 1

   Expanding brackets

   Single bracket: a(b + c) = ab + ac

   Double brackets: (a + b)(c + d) = ac + ad + bc + bd. Multiply every term in the first bracket by every term in the second.

   Example: (x + 3)(x − 5) = x^2 − 5x + 3x − 15 = x^2 − 2x − 15

2. 2

   Special products

   Perfect square: (a + b)^2 = a^2 + 2ab + b^2, and (a − b)^2 = a^2 − 2ab + b^2

   Difference of two squares: a^2 − b^2 = (a + b)(a − b)

   A common mistake is writing (a + b)^2 = a^2 + b^2. The middle term 2ab is always there.

3. 3

   Factorising

   Common factor: take out the highest common factor, e.g. 6x^2 + 9x = 3x(2x + 3).

   Quadratic x^2 + bx + c: find two numbers that multiply to c and add to b, e.g. x^2 + 7x + 12 = (x + 3)(x + 4).

   Check by expanding your answer.

4. 4

   Solving quadratics

   For ax^2 + bx + c = 0, first rearrange so one side is zero.

   • Factorise and set each bracket equal to zero.
   • Or use the quadratic formula: x = (−b ± √(b^2 − 4ac)) ÷ 2a.
   • Or complete the square.
5. 5

   Completing the square and the discriminant

   Completing the square: x^2 + bx + c = (x + b/2)^2 − (b/2)^2 + c. The turning point of y = (x + p)^2 + q is at (−p, q).

   Discriminant b^2 − 4ac: greater than 0 gives two real roots, equal to 0 gives one repeated root, less than 0 gives no real roots.

6. 6

   Straight lines

   Equation: y = mx + c, where m is the gradient and c is the y-intercept.

   Gradient between (x₁, y₁) and (x₂, y₂): m = (y₂ − y₁) ÷ (x₂ − x₁).

   Through a point: y − y₁ = m(x − x₁).

   Parallel lines have equal gradients. Perpendicular gradients multiply to −1.

7. 7

   Midpoint and distance

   Midpoint of (x₁, y₁) and (x₂, y₂): ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

   Distance between them: √((x₂ − x₁)^2 + (y₂ − y₁)^2), which comes from Pythagoras' theorem.

8. 8

   Laws of indices

   • a^m × a^n = a^(m + n)
   • a^m ÷ a^n = a^(m − n)
   • (a^m)^n = a^(mn)
   • a^0 = 1 (for a not equal to 0)
   • a^(−n) = 1 ÷ a^n
   • a^(1/n) is the nth root of a, and a^(m/n) is the nth root of a, raised to the power m
9. 9

   Laws of logarithms

   log_a(x) = y means a^y = x.

   • log(xy) = log x + log y
   • log(x ÷ y) = log x − log y
   • log(x^n) = n log x
   • log_a(a) = 1 and log_a(1) = 0
10. 10

    Simultaneous equations

    Elimination: multiply the equations so one variable has the same coefficient, then add or subtract to remove it.

    Substitution: make one variable the subject of one equation and substitute it into the other. Use this when one equation is quadratic.

    Substitute your answers back in to check.

11. 11

    Sequences

    Arithmetic sequence: nth term = a + (n − 1)d, where a is the first term and d the common difference. Sum of n terms = n/2 × (2a + (n − 1)d).

    Geometric sequence: nth term = ar^(n − 1), where r is the common ratio.

12. 12

    Rearranging formulas and inequalities

    Rearranging: do the same operation to both sides, undoing operations in reverse order until the letter you want is on its own.

    Inequalities: solve like equations, but when you multiply or divide both sides by a negative number, reverse the inequality sign.

    Example: −2x greater than 6 gives x less than −3.