[size=150]Select [b]all [/b]expressions that are perfect squares. Explain how you know.[/size][br][br][math](x+5)(5+x)[/math]

[math](x+5)(x-5)[/math]

[math](x-3)^2[/math]

[math]x-3^2[/math]

[math]x^2+8x+16[/math]

[math]x^2+10x+20[/math]

[size=150]One technique for solving quadratic equations is called [b]completing the square[/b][i].[/i] Here are two examples of how Diego and Mai completed the square to solve the same equation. [br][br][table][tr][td]Diego:[/td][td]Mai:[/td][/tr][tr][td][math]\displaystyle \begin {align} x^2+10x+9 &=0 \\x^2+10x &= \text-9 \\ x^2+10x+25 &=\text-9 + 25\\x^2+10x+25 &=16 \\ (x+5)^2 &=16\\ x+5=4 \quad & \text{or} \quad x+5=\text-4\\ x=\text-1 \quad & \text{or} \quad x=\text-9 \end{align}[/math][/td][td][math]\begin {align} x^2 + 10x + 9 &= 0\\ x^2 + 10x + 9 + 16 &= 16\\ x^2+10x+25 &=16\\ (x+5)^2&=16\\ x+5=4 \quad & \text{or} \quad x+5=\text-4\\ x=\text-1 \quad & \text{or} \quad x=\text-9 \end {align}[/math][br][/td][/tr][/table][br][/size][size=150][br]Study the worked examples. Then, try solving these equations by completing the square:[br][/size][br][math]x^2+6x+8=0[/math]

[math]x^2+12x=13[/math]

[math]0=x^2-10x+21[/math]

[math]x^2-2x+3=83[/math]

[math]x^2+40=14x[/math]

[size=150]Its total area is [math]x^2+35x[/math] square units.[br][br][/size]What is the length of the unlabeled side of each of the two rectangles?[br]

If we add lines to make the figure a square, what is the area of the entire figure?[br]

How is the process of finding the area of the entire figure like the process of building perfect squares for expressions like [math]x^2+bx[/math]?