The identities work for numbers as well as letters. Write an awkward number as a round number plus or minus a small number, then use an identity so that only easy multiplications remain. Not drawn to scale. Why: 652 is the area of a square of side 65=60+5. It splits into 602=3600, two rectangles of 60 x 5=300 and 52=25. So 652=3600+600+25=4225. This is (a+b)2=a2+2ab+b2 with a=60 and b=5. **Calculation** | **Identity** | **Choice of a and b** 512 | (a+b)2=a2+2ab+b2 | a=50, b=1 982 | (a-b)2=a2-2ab+b2 | a=100, b=2 46 x 54 | (a-b)(a+b)=a2-b2 | a=50, b=4 832-172 | a2-b2=(a+b)(a-b) | a=83, b=17 3022-4(302)+4 | a2-2ab+b2=(a-b)2 | a=302, b=2 Method - Choose a round number a close to the given numbers, such as a multiple of 10, 100 or 1000. - For a square, write the number as a+b or a-b. For a product of two numbers the same distance either side of a, write it as (a-b)(a+b). - For a difference of two squares, write it as (a+b)(a-b); no squaring is needed. - For three terms that are a square, twice a product and a square, read them as a2+-2ab+b2 and write (a+- b)2. - Work out the easy products and combine them. Example 982 = (100-2)2 = 1002-2(100)(2)+22 = 10,000-400+4 = 9604 46 x 54 = (50-4)(50+4) = 502-42 = 2500-16 = 2484 832-172 = (83+17)(83-17) = 100 x 66 = 6600 3022-4(302)+4 = 3022-2(302)(2)+22 = (302-2)2 = 3002=90,000 When part (a) asks for an expansion and part (b) says "Hence", choose the value of the letter that turns the expansion into the numbers. For example, (3x+2)(3x-2)=9x2-4. Taking 3x=600, so x=200, gives 602 x 598=9(200)2-4=359,996. Check: estimate first. 982 is a little less than 1002=10,000, so 9604 is sensible. A common error is (100-2)2=10,000-4, which drops the middle term 2(100)(2)=400.

Evaluating numbers with identities

Secondary 2

The identities work for numbers as well as letters. Write an awkward number as a round number plus or minus a small number, then use an identity so that only easy multiplications remain.

Not drawn to scale.

Why: $65^2$ is the area of a square of side $65=60+5$. It splits into $60^2=3600$, two rectangles of $60\times5=300$ and $5^2=25$. So $65^2=3600+600+25=4225$. This is $(a+b)^2=a^2+2ab+b^2$ with $a=60$ and $b=5$.

Calculation	Identity	Choice of $a$ and $b$
$51^2$	$(a+b)^2=a^2+2ab+b^2$	$a=50,\ b=1$
$98^2$	$(a-b)^2=a^2-2ab+b^2$	$a=100,\ b=2$
$46\times54$	$(a-b)(a+b)=a^2-b^2$	$a=50,\ b=4$
$83^2-17^2$	$a^2-b^2=(a+b)(a-b)$	$a=83,\ b=17$
$302^2-4(302)+4$	$a^2-2ab+b^2=(a-b)^2$	$a=302,\ b=2$

Method

Choose a round number $a$ close to the given numbers, such as a multiple of $10$, $100$ or $1000$.
For a square, write the number as $a+b$ or $a-b$. For a product of two numbers the same distance either side of $a$, write it as $(a-b)(a+b)$.
For a difference of two squares, write it as $(a+b)(a-b)$; no squaring is needed.
For three terms that are a square, twice a product and a square, read them as $a^2\pm2ab+b^2$ and write $(a\pm b)^2$.
Work out the easy products and combine them.

Example

$\begin{aligned} 98^2 &= (100-2)^2 \\ &= 100^2-2(100)(2)+2^2 \\ &= 10\,000-400+4 \\ &= 9604 \end{aligned}$

$\begin{aligned} 46\times54 &= (50-4)(50+4) \\ &= 50^2-4^2 \\ &= 2500-16 \\ &= 2484 \end{aligned}$

$\begin{aligned} 83^2-17^2 &= (83+17)(83-17) \\ &= 100\times66 \\ &= 6600 \end{aligned}$

$\begin{aligned} 302^2-4(302)+4 &= 302^2-2(302)(2)+2^2 \\ &= (302-2)^2 \\ &= 300^2=90\,000 \end{aligned}$

When part (a) asks for an expansion and part (b) says "Hence", choose the value of the letter that turns the expansion into the numbers. For example, $(3x+2)(3x-2)=9x^2-4$. Taking $3x=600$, so $x=200$, gives $602\times598=9(200)^2-4=359\,996$.

Check: estimate first. $98^2$ is a little less than $100^2=10\,000$, so $9604$ is sensible. A common error is $(100-2)^2=10\,000-4$, which drops the middle term $2(100)(2)=400$.

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Calculation	Identity	Choice of \(a\) and \(b\)
\(51^2\)	\((a+b)^2=a^2+2ab+b^2\)	\(a=50,\ b=1\)
\(98^2\)	\((a-b)^2=a^2-2ab+b^2\)	\(a=100,\ b=2\)
\(46\times54\)	\((a-b)(a+b)=a^2-b^2\)	\(a=50,\ b=4\)
\(83^2-17^2\)	\(a^2-b^2=(a+b)(a-b)\)	\(a=83,\ b=17\)
\(302^2-4(302)+4\)	\(a^2-2ab+b^2=(a-b)^2\)	\(a=302,\ b=2\)