Squaring numbers

Algebra Level 2

Numbers could be squared using the following algorithm:

$\large {41 \times 41=\color{#3D99F6}40 \times \color{#D61F06}42 + \color{#69047E}1^2=\color{#20A900}1681 \\ 32 \times 32=\color{#3D99F6}30 \times \color{#D61F06}34 + \color{#69047E}2^2=\color{#20A900}1024 \\ 57 \times 57=\color{#3D99F6}54 \times \color{#D61F06}60 + \color{#69047E}3^2=\color{#20A900}3249 \\ 89 \times 89=\color{#3D99F6}88 \times \color{#D61F06}90 +\color{#69047E}1^2=\color{#20A900}7921 \\ \vdots }$

Find $78 \times 78$

The answer is 6084.

3 solutions

Mrigank Shekhar Pathak
Nov 3, 2017

Here we are using the following identity:

$a^2-b^2=(a+b)\cdot(a-b) \\ \therefore a^2=(a+b)\cdot(a-b)+b^2$

It would be useful if adjust $b$ in such a way that either $a+b$ or $a-b$ have units digit $0$ which makes multiplication easier

hence, putting $a=78,b=2$ we get

$\large 78 \times 78=\color{#3D99F6}76 \times \color{#D61F06}80 +\color{#69047E}2^2=\color{#20A900}6084$

Very useful,thanks.I think it is a great explanation.

Edward Christian - 1 year, 10 months ago

Munem Shahriar
Nov 3, 2017

$(78)^2$

$= (76 + 2)^2$

$= 76^2 + 2 \times 76 \times 2 + 2^2$

$= 5776 + 304 + 4$

$= 6084$

Edwin Gray
Jan 25, 2019

78x78=(78+3)x(78-3)+3^2=81x75+9=8100x(3/4)+9 =6075+9=6084. Ed Gray

Squaring numbers

The answer is 6084.

3 solutions

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