Question

Explain the polynomial geometrically: $(6x-5{)}^2=36x^2+5^2-60x$

Answer 1

Step 1: Draw a square $ACDF$ with $AC=6x$.
Step 2: Cut $AB=5$, so that $BC=(6x-5)$.
Step 3: Complete the squares and rectangle as shown in the diagram.
Step 4: Area of yellow square $IDEO=$ Area of square $ACDF-$ Area of rectangle $GOFE-$ Area of rectangle $BCIO-$ Area of red square $ABOG$
Therefore, $(6x-5{)}^2=(6x{)}^2-5(6x-5)-5(6x-5)-(5{)}^2$
$=$ $36x^2-30x+5^2-30x+5^2-5^2$
$=$ $36x^2+5^2-60x$
Hence, geometrically we proved the identity $(6x-5{)}^2=36x^2+5^2-60x$.