Advanced Patterns and Square Root Techniques

Theory: Square of a number \(a + b\) can be found using identity:

$$ (a + b)^2 = a^2 + 2ab + b^2 $$

Example:

Question: Find the square of 23 using identity.

Solution: $$ 23 = 20 + 3 \\ 23^2 = 20^2 + 2 \cdot 20 \cdot 3 + 3^2 = 400 + 120 + 9 = 529 $$

Theory: The square of a number ending in 5 follows this pattern:

$$ (10a + 5)^2 = 100a(a + 1) + 25 $$

Example:

Question: Find \(65^2\)

Solution: $$ a = 6,\quad 100 \cdot 6 \cdot 7 + 25 = 4200 + 25 = 4225 $$

Theory: For any natural number \(m > 1\), a triplet is given by:

$$ \text{Triplet: } (2m,\ m^2 – 1,\ m^2 + 1) $$

Example:

Question: Find a Pythagorean triplet where the smallest number is 8.

Solution:$$ 2m = 8 \Rightarrow m = 4 \\ m^2 – 1 = 15,\quad m^2 + 1 = 17 \\ \text{Triplet: } 8,\ 15,\ 17 $$

Theory: Square root is the inverse operation of squaring.

$$ \sqrt{n^2} = n $$

Example:

Question: What is the square root of 144?

Solution: $$ \sqrt{144} = 12 $$

Theory: A perfect square can be reduced to 0 by subtracting successive odd numbers starting from 1. The number of steps gives the square root.

Example:

Question: Find the square root of 81 using subtraction.

Solution: $$ 81 – 1 = 80 \\ 80 – 3 = 77 \\ 77 – 5 = 72 \\ \dots \rightarrow 17 – 17 = 0 \\ \text{9 steps} \Rightarrow \sqrt{81} = 9 $$

Theory: Pair the prime factors and take one from each pair.

Example:

Question: Find \(\sqrt{324}\)

Solution: $$ 324 = 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \\ = (2 \cdot 3 \cdot 3)^2 = 18^2 \Rightarrow \sqrt{324} = 18 $$

Theory: Use digit grouping in pairs from right, find digits step-by-step using division and doubling method.

Example:

Question: Find \(\sqrt{4096}\)

Solution: $$ \text{Group: } 40\ 96 \\ 6^2 = 36 < 40 \Rightarrow 6 \\ \text{Bring down 96, double 6 = 12} \\ \text{Find } x \text{ such that } 12x \cdot x \leq 496 \Rightarrow x = 4 \\ \text{So, root = } 64 $$

Theory: Multiply by the missing unpaired prime factors to complete squares.

Example:

Question: Find the smallest number to multiply 48 to get a perfect square.

Solution: $$ 48 = 2^4 \cdot 3 \Rightarrow \text{Unpaired: 3} \\ 48 \cdot 3 = 144 = 12^2 $$

Theory: Divide the number by the unpaired factor to remove imperfection.

Example:

Question: Find smallest number to divide 90 to get a perfect square.

Solution: $$ 90 = 2 \cdot 3^2 \cdot 5 \\ \text{Try: } 90 \div 5 = 18, 90 \div 2 = 45 \text{ (none perfect)} \\ 90 \div 10 = 9 = 3^2 \Rightarrow \text{Divide by } 10 $$

Theory: Word problems often ask for side length from given area or number of elements in square form.

Example:

Question: A square plot has area 2304 m². Find its side.

Solution: $$ \sqrt{2304} = 48 \Rightarrow \text{Side = 48 m} $$

Theory: Group decimal digits in pairs after the decimal point and apply long division.

Example:

Question: Find \(\sqrt{17.64}\)

Solution: $$ \text{Group: } 17.64 \rightarrow 17\ \overline{64} \\ \sqrt{17.64} = 4.2 $$

Theory: If a number has \(n\) digits, then digits in its square root is:
– \(\frac{n}{2}\) if even
– \(\frac{n+1}{2}\) if odd

Example:

Question: Estimate digits in \(\sqrt{390625}\)

Solution: $$ 390625 \text{ has 6 digits} \Rightarrow \text{Root has } 3 \text{ digits} $$

Theory: In a square of side \(a\), diagonal is \(\sqrt{2}a\). Use Pythagoras in right triangles to find unknown sides.

Example:

Question: Find hypotenuse if legs are 6 cm and 8 cm.

Solution: $$ AC = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\ \text{cm} $$