Question

1. competency check replace each unknown box, ■, with algebra tiles tomake a true statement. explain your reasoning.a)

Explanation:

Step1: Analyze the left - hand side and right - hand side of the equation in terms of algebra tiles

Let's assume the large square is \(x^{2}\), the rectangle is \(x\), and the small square is \(1\).
The left - hand side has \(2x^{2}+2x - 2\) (two large squares, two rectangles, and two red small squares which represent \(- 1\) each). The right - hand side has \(x^{2}+3x + 1\) (one large square, three rectangles, and one white small square which represents \(+1\)).

Step2: Set up the equation

Let the unknown box be \(ax^{2}+bx + c\). Then \((2x^{2}+2x - 2)+(ax^{2}+bx + c)=x^{2}+3x + 1\).
By combining like terms: \((2 + a)x^{2}+(2 + b)x+(c - 2)=x^{2}+3x + 1\).

Step3: Solve for \(a\), \(b\), and \(c\)

For the \(x^{2}\) terms: \(2 + a=1\), so \(a=-1\).
For the \(x\) terms: \(2 + b=3\), so \(b = 1\).
For the constant terms: \(c-2 = 1\), so \(c=3\).

Answer:

The unknown box should have \(-x^{2}+x + 3\) (one negative large square, one positive rectangle, and three positive small squares). The reasoning is based on equating the coefficients of the like - terms (\(x^{2}\), \(x\), and the constant term) on both sides of the equation formed by the sum of the left - hand side and the unknown box equal to the right - hand side.