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1. Tema: Factorising quadratics

   1. Factorise x² + 5x + 6.

   A(x + 1)(x + 6)B(x + 2)(x + 3)C(x − 2)(x − 3)D(x + 5)(x + 1)
   Solução

   B. (x + 2)(x + 3)

   • Find two numbers that multiply to 6 and add to 5: 2 and 3.
   • x² + 5x + 6 = (x + 2)(x + 3).

2. Tema: Quadratic equations

   2. Solve x² − 9 = 0.

   Ax = 3 or x = −3Bx = 3 onlyCx = 9 or x = −9Dx = 4.5
   Solução

   A. x = 3 or x = −3

   • x² = 9.
   • x = √9 = 3 or x = −3. A quadratic usually has two solutions.

3. Tema: Quadratic equations

   3. Solve 2x² + 3x − 2 = 0.

   Ax = −1/2 or x = 2Bx = 1 or x = −2Cx = 2 or x = −1Dx = 1/2 or x = −2
   Solução

   D. x = 1/2 or x = −2

   • Factorise: (2x − 1)(x + 2) = 0.
   • 2x − 1 = 0 gives x = 1/2, and x + 2 = 0 gives x = −2.

4. Tema: Simultaneous equations

   4. Solve 2x + y = 7 and x − y = 2.

   Ax = 2, y = 3Bx = 3, y = −1Cx = 3, y = 1Dx = 1, y = 5
   Solução

   C. x = 3, y = 1

   • Add the equations to eliminate y: 3x = 9, so x = 3.
   • Substitute: 3 − y = 2, so y = 1.

5. Tema: Expanding brackets

   5. Expand (x + 4)(x − 3).

   Ax² − x − 12Bx² + x − 12Cx² + 7x − 12Dx² + x + 12
   Solução

   B. x² + x − 12

   • x × x = x², x × −3 = −3x, 4 × x = 4x, 4 × −3 = −12.
   • x² − 3x + 4x − 12 = x² + x − 12.

6. Tema: Algebraic fractions

   6. Simplify (x² − 16)/(x + 4).

   Ax − 4Bx + 4Cx − 16Dx² − 4
   Solução

   A. x − 4

   • x² − 16 is a difference of two squares: (x + 4)(x − 4).
   • Cancel (x + 4): x − 4.

7. Tema: Completing the square

   7. Write x² + 6x + 1 in the form (x + a)² + b.

   A(x + 3)² + 1B(x + 6)² − 35C(x + 3)² − 10D(x + 3)² − 8
   Solução

   D. (x + 3)² − 8

   • Halve the x coefficient: (x + 3)² = x² + 6x + 9.
   • x² + 6x + 1 = (x + 3)² − 9 + 1 = (x + 3)² − 8.

8. Tema: Composite functions

   8. f(x) = 3x − 1 and g(x) = x². Find fg(2).

   A25B12C11D7
   Solução

   C. 11

   • fg(2) means f(g(2)). Work out g first: g(2) = 4.
   • f(4) = 3 × 4 − 1 = 11.

9. Tema: Inverse functions

   9. f(x) = 2x + 5. Find f⁻¹(x).

   A(x + 5)/2B(x − 5)/2C2x − 5D1/(2x + 5)
   Solução

   B. (x − 5)/2

   • Write y = 2x + 5 and make x the subject: x = (y − 5)/2.
   • So f⁻¹(x) = (x − 5)/2.

10. Tema: Quadratic sequences

    10. Find the nth term of 2, 5, 10, 17, ...

    An² + 1B3n − 1Cn² + 2D2n² − 1
    Solução

    A. n² + 1

    • Differences: 3, 5, 7. Second differences: 2, so the sequence starts n².
    • n² gives 1, 4, 9, 16. Each term is 1 more, so n² + 1.