Question

Expand the identity geometrically: $(x+a)(x+b)$

Answer 1

Step 1: Draw a square and cut into $4$ parts.
Step 2: There are $4$ rectangle and $1$ square.
Step 3: Area of the full rectangle, $ABCD=$ $(x+a)(x+b)$
Step 4: Now we have to find the area of inside square and rectangle as shown in the figure.
Step 5: Consider the area of pink square $=$ $x^2$ and the area of yellow rectangle $=$ length $\times$ breadth $=$ $bx$
Step 6: Area of blue rectangle $=$ $ax$ and the area of green rectangle $=$ $ab$
Step 7: Area of full rectangle $=$ area of pink square $+$ area of yellow rectangle $+$ area of blue rectangle $+$ area of green rectangle.
i.e., $(x+a)(x+b)=x^2+bx+ax+ab$
$(x+a)(x+b)=x^2+x(a+b)+ab$
Hence, geometrically we proved the identity $(x+a)(x+b)=x^2+x(a+b)+ab$.