Question

Chad was asked whether the following equation is an identity (3x+2y)^2=(3x+2y)(3x-2y)+2(2y)^2 He performed the following steps: (3x+2y)^2 1}9x^2+4y^2 {Step }2} 9x^2+4y^2+4y^2-4y^2 {Step }3} (9x^2-4y^2)+8y^2 {Step }4}=(3x+2y)(3x-2y)+2(2y)^2 For this reason, Chad stated that the equation is a true identity. Is Chad correct? If not, in which step did he make a mistake?

Answer 1

Chad made a mistake in Step 1 by not expanding the left side of the equation correctly, omitting the 12xy term. The correct expansion of (3x+2y)^2 is 9x^2 + 12xy + 4y^2, which makes the equation not an identity as the two sides are not equal.

Step-by-step explanation:

The student Chad is checking if the equation (3x+2y)^2=(3x+2y)(3x-2y)+2(2y)^2 is an identity. To verify this, he expanded the left side and modified the right side but he made a mistake in his calculations. Let's correct it.

• Step 1: Expand the left side using the FOIL method (First, Outside, Inside, Last). (3x+2y)^2 = 9x^2 + 6xy + 6xy + 4y^2 = 9x^2 + 12xy + 4y^2. Chad incorrectly expanded to only 9x^2 + 4y^2.
• Step 2: Expand the right side as well. (3x+2y)(3x-2y) + 2(2y)^2 = (9x^2 - 4y^2) + 2(4y^2) = 9x^2 - 4y^2 + 8y^2.
• Step 3: Compare the two sides. We have 9x^2 + 12xy + 4y^2 on the left and 9x^2 + 4y^2 on the right. Clearly, these are not the same because the left side has an additional 12xy.

Therefore, Chad's statement that the equation is a true identity is incorrect, and he made a mistake in Step 1 by not expanding the left side correctly.