To determine the value of \( x \) that Eli added to each side of the cement slab, we can begin by defining the dimensions of the new slab.
Initially, the dimensions of the slab are:
- Length = 8 ft
- Width = 6 ft
When Eli adds \( x \) feet to each side, the new dimensions become:
- New Length = \( 8 + 2x \) (adding \( x \) to each side of the length)
- New Width = \( 6 + 2x \) (adding \( x \) to each side of the width)
The area of the new slab is given as 120 square feet. Thus, we can set up the equation for the area of the new slab:
\[ (8 + 2x)(6 + 2x) = 120 \]
Next, we can expand this equation:
\[ 48 + 16x + 12x + 4x^2 = 120 \]
Combining like terms yields:
\[ 4x^2 + 28x + 48 = 120 \]
Now, we can simplify this equation by moving 120 to the left side:
\[ 4x^2 + 28x + 48 - 120 = 0 \]
This simplifies to:
\[ 4x^2 + 28x - 72 = 0 \]
Next, we can divide the entire equation by 4 to simplify further:
\[ x^2 + 7x - 18 = 0 \]
Now we can factor this quadratic equation. We need to find two numbers that multiply to \(-18\) and add to \(7\). These numbers are \(9\) and \(-2\). Thus, we can factor the equation as follows:
\[ (x + 9)(x - 2) = 0 \]
Setting each factor equal to zero gives us the possible solutions for \( x \):
\[ x + 9 = 0 \quad \Rightarrow \quad x = -9 \quad \text{(not a valid solution since dimensions can't be negative)} \] \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
Since \( x \) cannot be negative, the only valid solution is:
\[ x = 2 \]
Therefore, Eli added \( \boxed{2} \) feet to each side of the cement slab.