Square Roots Without a Calculator

Algebra Level 2

20 ∗ 21 ∗ 22 ∗ 23 + 1 = ? \sqrt{20*21*22*23 + 1} = \ ?

NSF Math BEE Finals
441 461 459 449

Rmflute Shrivastav by Rmflute Shrivastav , 19, USA

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3 solutions

Here is my basic way: factoring. Let 20 = A 20 =A

A ( A + 1 ) ( A + 2 ) ( A + 3 ) + 1 \sqrt{A(A+1)(A+2)(A+3)+ 1}

= A 4 + 6 A 3 + 11 A 2 + 6 A + 1 =\sqrt{ A^4+6A^3+11A^2+6A+1}

= ( A 2 + 3 A + 1 ) 2 =\sqrt{(A^2+3A+1)^2}

⇒ A 2 + 3 A + 1 \Rightarrow A^2+3A+1

Purge A = 20 A=20 :

2 0 2 + 3 × 20 + 1 = 461 20^2+3\times 20+1 = \boxed{461}

12 Helpful 0 Interesting 0 Brilliant 0 Confused

From my point of view, I would prefer to put the A and it's equation first, THEN put A = 20, but that's just my preference. It's still fine if the one you state is your preference.

Gian Sanjaya - 5 years, 9 months ago

Great Solution, easy to understand and simple! How did you get from step 2 to step 3?

Rmflute Shrivastav - 5 years, 9 months ago

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A 4 + 6 A 3 + 11 A 2 + 6 A + 1 A^4+6A^3+11A^2+6A+1

= A 4 + 3 A 3 + A 2 + 3 A 3 + 9 A 2 + 3 A + 3 A 2 + 3 A + 1 4 = \color{#D61F06}{A^4}+3A^3+A^2+3A^3+\color{#D61F06}{9A^2}+3A+3A^2+3A+\color{#D61F06}{1^4}

= ( A 2 + 3 A + 1 ) 2 =(A^2+3A+1)^2

Adam Phúc Nguyễn - 5 years, 9 months ago
Ishita .S
Aug 28, 2015

20 * 21 22 23+1 = 212521 Root of 212521 is 461

0 Helpful 0 Interesting 0 Brilliant 1 Confused

Without a calculator?

Tim J. - 3 years, 11 months ago
Antonio Rangel
Aug 28, 2015

20 * 21 * 22 * 23 + 1 = (20*23+1)^2

x(x+1)(x+2)(x+3)+1 = (x^2+3x+1)^2

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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