Arithmetic and terms

Binomial formulas

The three binomial formulas save time when you expand brackets and are the key to factorising and completing the square. German tests call them binomische Formeln.

The rules

• First: (a + b)² = a² + 2ab + b².
• Second: (a − b)² = a² − 2ab + b².
• Third: (a + b)(a − b) = a² − b².
• Read them backwards to factorise: x² − 9 = (x + 3)(x − 3).

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. Use (x + 7)² = x² + 14x + 49 to evaluate it at x = 2.

   • a18
   • b81
   • c53
   • d32
   Show solution

   Answer: 81

   (x + a)² = x² + 2ax + a².

   2² + 14·2 + 49 = 4 + 28 + 49 = 81.

2. 2. Use (x + 6)² = x² + 12x + 36 to evaluate it at x = 3.

   • a18
   • b81
   • c45
   • d87
   Show solution

   Answer: 81

   (x + a)² = x² + 2ax + a².

   3² + 12·3 + 36 = 9 + 36 + 36 = 81.

3. 3. Expand (x + 6)². What is the coefficient of x?

   • a12
   • b6
   • c13
   • d36
   Show solution

   Answer: 12

   (x + a)² = x² + 2ax + a².

   The middle term is 2·a·x = 12x.

4. 4. Use (x + 5)² = x² + 10x + 25 to evaluate it at x = 4.

   • a81
   • b56
   • c18
   • d41
   Show solution

   Answer: 81

   (x + a)² = x² + 2ax + a².

   4² + 10·4 + 25 = 16 + 40 + 25 = 81.

5. 5. Expand (x + 3)². What is the coefficient of x?

   • a7
   • b3
   • c9
   • d6
   Show solution

   Answer: 6

   (x + a)² = x² + 2ax + a².

   The middle term is 2·a·x = 6x.

6. 6. Use (x + 5)² = x² + 10x + 25 to evaluate it at x = 3.

   • a16
   • b64
   • c34
   • d39
   Show solution

   Answer: 64

   (x + a)² = x² + 2ax + a².

   3² + 10·3 + 25 = 9 + 30 + 25 = 64.

Common mistakes

• Dropping the middle term: (x + 3)² is x² + 6x + 9, not x² + 9.
• Getting the sign of the middle term wrong in the second formula.
• Not spotting a² − b² inside a fraction you could simplify.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs