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(a-b)^2 Expansion and Visualisation

Geometrically...

Question

Geometrically explain the polynomial: $(2 x - 3)^{2} = 4 x^{2} + 3^{2} - 12 x$

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Solution

Step 1: Draw a square $A C D F$ with $A C = 2 x$ .
Step 2: Cut $A B = 3$ so that $B C = (2 x - 3)$ .
Step 3: Complete the squares and rectangle as shown in the diagram.
Step 4: Area of yellow square $I D E O =$ Area of square $A C D F -$ Area of rectangle $G O F E -$ Area of rectangle $B C I O -$ Area of red square $A B O G$
Therefore, $(2 x - 3)^{2} = (2 x)^{2} - 3 (2 x - 3) - 3 (2 x - 3) - (3)^{2}$
$=$ $4 x^{2} - 6 x + 3^{2} - 6 x + 3^{2} - 3^{2}$
$=$ $4 x^{2} + 3^{2} - 12 x$
Hence, geometrically we proved the identity $(2 x - 3)^{2} = 4 x^{2} + 3^{2} - 12 x$ .

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Similar questions

Q. Geometrically explain the polynomial:

(2 x + 3)^{2} = 4 x^{2} + 3^{2} + 12 x

Q. Which of the following figure is true for the polynomial

4 x^{2} + 3^{2} - 12 x

?

\frac{(4 x + 1)^{2} + (2 x + 3)^{2}}{4 x^{2} + 12 x + 9} = \frac{61}{36}

Q. Solve the following equation:

\frac{(4 x + 1)^{2} + (2 x + 3)^{2}}{4 x^{2} + 12 x + 9} = \frac{61}{36}

(2 x - 3)^{2} = 4 x^{2} - 6 x + 9

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