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The short multiplication formulas

Algebra, grade 9 Lesson 5 of 26Learning

The short multiplication formulas

The short multiplication formulas are three products that come up so often that they are worth knowing by heart:

The square of a sum:

(a+b)2\displaystyle (a+b)^2 =\displaystyle = a2+2ab+b2\displaystyle a^2+2ab+b^2

The square of a difference:

(a−b)2\displaystyle (a-b)^2 =\displaystyle = a2−2ab+b2\displaystyle a^2-2ab+b^2

The difference of squares:

(a+b)(a−b)\displaystyle (a+b)(a-b) =\displaystyle = a2−b2\displaystyle a^2-b^2

Each one comes from the extended distributive law.

The square of a sum

The square of a sum is the square of the first term, plus twice their product, plus the square of the second:

(a+b)2=(a+b)(a+b)=a2+ab+ba+b2=a2+2ab+b2(a+b)^2=(a+b)(a+b)=a^2+ab+ba+b^2=a^2+2ab+b^2
  • (x+3)2=x2+2⋅x⋅3+32=x2+6x+9\displaystyle {(x+3)^2}={x^2+2\cdot x\cdot3+3^2}={x^2+6x+9}
  • (2y+5)2=4y2+20y+25\displaystyle {(2y+5)^2}={4y^2+20y+25}
The square of a sum in areas
(a+b)2\displaystyle (a+b)^2 is not a2+b2\displaystyle a^2+b^2: (3+4)2=49\displaystyle {(3+4)^2}={49}, but 32+42=25\displaystyle {3^2+4^2}={25}. The missing part is 2ab\displaystyle 2ab – the two rectangles.

A square with side a+b\displaystyle a+b is cut into four parts: a square a2\displaystyle a^2, a square b2\displaystyle b^2 and two rectangles ab\displaystyle ab. ababa2ababb2 So (a+b)2=a2+2ab+b2\displaystyle {(a+b)^2}={a^2+2ab+b^2}.

The square of a difference
The last term is always plus: (−b)2=+b2\displaystyle {(-b)^2}={+b^2}.

The square of a difference is the same, with a minus before the middle term:

(a−b)2=(a−b)(a−b)=a2−ab−ba+b2=a2−2ab+b2(a-b)^2=(a-b)(a-b)=a^2-ab-ba+b^2=a^2-2ab+b^2
  • (x−5)2=x2−10x+25\displaystyle {(x-5)^2}={x^2-10x+25}
  • (3a−1)2=9a2−6a+1\displaystyle {(3a-1)^2}={9a^2-6a+1}
Exercise 1

Open the brackets with the formulas.

a

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        d

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          e

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            The difference of squares

            The product of a sum and a difference of the same two terms is the difference of their squares – the middle terms cancel out:

            (a+b)(a−b)=a2−ab+ab−b2=a2−b2(a+b)(a-b)=a^2-ab+ab-b^2=a^2-b^2
            • (x+7)(x−7)=x2−49\displaystyle {(x+7)(x-7)}={x^2-49}
            • (3x−2)(3x+2)=9x2−4\displaystyle {(3x-2)(3x+2)}={9x^2-4}
            Exercise 2

            Open the brackets with the formula.

            a

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              b

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                c

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  d

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                    Mental arithmetic

                    The formulas turn hard products into easy ones: write each number as a round number plus or minus a little.

                    • 512=(50+1)2=2500+100+1=2601\displaystyle {51^2}={(50+1)^2}={2500+100+1}={2601}
                    • 982=(100−2)2=10000−400+4=9604\displaystyle {98^2}={(100-2)^2}={10000-400+4}={9604}
                    • 49⋅51=(50−1)(50+1)=2500−1=2499\displaystyle {49\cdot51}={(50-1)(50+1)}={2500-1}={2499}
                    Exercise 3

                    Calculate without a calculator, with a formula.
                    Start by writing each number as a round number plus or minus a little.

                    a

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                      b

                        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                        c

                          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                          d

                            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                            Simplifying with the formulas
                            A minus before a square: keep the result in brackets until the signs are flipped.

                            Open every square and product with its formula, then collect like terms:

                            (x+3)2−(x−3)2=x2+6x+9−(x2−6x+9)=x2+6x+9−x2+6x−9=12x(x+3)^2-(x-3)^2=x^2+6x+9-(x^2-6x+9)=x^2+6x+9-x^2+6x-9=12x
                            Exercise 4

                            Simplify.

                            a

                              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                              b

                                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                                c

                                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                                  d

                                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                                    Exercise 5

                                    Ron wrote: (x+6)2=x2+36\displaystyle {(x+6)^2}={x^2+36}.
                                    Substitute x=1\displaystyle {x}={1} in both sides to show that he is wrong.
                                    Which parts of the square in the picture did he leave out?
                                    Open the brackets correctly.

                                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                                      Exercise 6

                                      What is missing from the boxes?
                                      (x+□)2=x2+□+49\displaystyle {(x+\square)^2}={x^2+\square+49}, (□−5)2=□−30y+25\displaystyle {\quad(\square-5)^2}={\square-30y+25}, (□+□)(□−□)=x2−64\displaystyle {\quad(\square+\square)(\square-\square)}={x^2-64}.