A square piece of paper has an area of $$ x^{2} $$ square units. A rectangular strip with a width of 2 units and a length of $$ x $$ units is cut off of the square piece of paper. The remaining piece of paper has an area of 120 $$ \begin{array}{l}x^{2}-2 x-120=0 \ ext { square units. } \ \begin{array}{ll}x^{2}+2 x-120=0 \ x^{2}-2 x+120=0 \ x^{2}+2 x+120=0\end{array}\end{array} $$

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A square piece of paper has an area of $$ x^{2} $$ square units. A rectangular strip with a width of 2 units and a length of $$ x $$ units is cut off of the square piece of paper. The remaining piece of paper has an area of 120 $$ \begin{array}{l}x^{2}-2 x-120=0 \ 	ext { square units. } \ \begin{array}{ll}x^{2}+2 x-120=0 \ x^{2}-2 x+120=0 \ x^{2}+2 x+120=0\end{array}\end{array} $$

UpStudy AI · Accepted Answer

**Step 1. Formulate the equation**

- The area of the square is given by $$ x^2 $$ square units.
- The area of the rectangular strip removed is its length times its width: $$ \text{Length} = x $$ and $$ \text{Width} = 2 $$, so the area is $$ 2x $$ square units.
- The remaining area of the paper is the area of the square minus the area of the strip:
  
  $$
  x^2 - 2x = 120.
  $$

- Rearranging the equation gives:

$$
  x^2 - 2x - 120 = 0.
  $$

**Step 2. Solve the quadratic equation**

We solve

$$
x^2 - 2x - 120 = 0.
$$

Factor the quadratic by finding two numbers that multiply to $$-120$$ and add to $$-2$$. The numbers $$10$$ and $$-12$$ work because:

$$
10 \times (-12) = -120 \quad \text{and} \quad 10 + (-12) = -2.
$$

Thus, we factor the equation as:

$$
(x + 10)(x - 12) = 0.
$$

Setting each factor equal to zero:

- $$x + 10 = 0 \quad \Longrightarrow \quad x = -10,$$
- $$x - 12 = 0 \quad \Longrightarrow \quad x = 12.$$

Since $$x$$ represents a length, it must be positive. Therefore, the solution is:

$$
x = 12.
$$

**Step 3. Verification**

- Area of the square: $$x^2 = 12^2 = 144$$ square units.
- Area of the rectangular strip: $$2x = 2 \times 12 = 24$$ square units.
- Remaining area: $$144 - 24 = 120$$ square units.

The answer is consistent with the problem statement.
$$ x = 12 $$

A square piece of paper has an area of square
units. A rectangular strip with a width of 2 units and a
length of units is cut off of the square piece of paper.
The remaining piece of paper has an area of 120

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A square piece of paper has an area of square units. A rectangular strip with a width of 2 units and a length of units is cut off of the square piece of paper. The remaining piece of paper has an area of 120