[size=150]Elena says, “[math]\left(x+3\right)^2[/math] can be expanded into [math]x^2+6x+9[/math]. Likewise, [math]\left(2x+3\right)^2[/math] can be expanded into [math]4x^2+6x+9[/math].”[/size][br][br]Find an error in Elena’s statement and correct the error. Show your reasoning.
[math]\left(4x+1\right)^2[/math]
[math]\left(5x-2\right)^2[/math]
[math]\left(\frac{1}{2}x+7\right)^2[/math]
[math]\left(3x+n\right)^2[/math]
[math]\left(kx+m\right)^2[/math]
Decide if [math]4x^2+12x+9[/math] is a perfect square. If so, write an equivalent expression of the form [math]\left(kx+m\right)^2[/math]. If not, suggest one change to turn it into a perfect square.[br]
Decide if [math]4x^2+8x+25[/math] is a perfect square. If so, write an equivalent expression of the form [math]\left(kx+m\right)^2[/math]. If not, suggest one change to turn it into a perfect square.[br][br]
[math]25x^2+40x=-12[/math]
[math]36x^2-60x+10=-6[/math]
[table][tr][td][size=150]Here are three methods for solving [math]3x^2+8x+5=0[/math]. [br]Try to make sense of each method.[/size][/td][td]Method 1:[/td][/tr][tr][td][/td][td][math]\displaystyle \begin {align}3x^2 + 8x + 5 &= 0\\ (3x + 5)(x + 1) &= 0 \end{align}[/math][br][math]\displaystyle \begin {align} x = \text- \frac53 \quad \text{or} \quad x = \text-1\end {align}[/math][br][/td][/tr][tr][td]Method 2:[/td][td]Method 3:[/td][/tr][tr][td][math]\displaystyle \begin {align} 3x^2 + 8x + 5 &= 0\\ 9x^2 + 24x + 15 &= 0\\ (3x)^2 + 8(3x) + 15 &= 0\\ U^2 + 8U + 15 &= 0\\ (U+5)(U+3) &= 0 \end{align}[/math][br][math]\displaystyle \begin {align} U = \text-5 \quad &\text{or} \quad U = \text-3\\3x = \text-5 \quad &\text{or} \quad 3x = \text-3\\ x = \text- \frac53 \quad &\text{or} \quad x = \text-1 \end{align}[/math][br][/td][td][math]\displaystyle \begin {align} 3x^2 + 8x + 5 &= 0\\ 9x^2 + 24x + 15 &= 0\\9x^2 + 24x + 16 &= 1\\(3x + 4)^2 &= 1 \end{align}[/math][br][math]\displaystyle \begin {align} 3x+4 = 1 \quad & \text{or} \quad 3x+4 = \text-1\\x = \text-1 \quad & \text{or} \quad x = \text- \frac53 \end {align}[/math][/td][/tr][/table][br][size=150]Once you understand the methods, use each method at least one time to solve these equations.[/size][br][br][math]5x^2+17x+6=0[/math]
[math]6x^2+19x=-10[/math]
[math]8x^2-33x+4=0[/math]
[math]8x^2-26x=-21[/math]
[math]10x^2+37x=36[/math]
[math]12x^2+20x-77=0[/math]
Find the solutions to [math]3x^2-6x+\frac{9}{4}=0[/math]. Explain your reasoning.