Find, algebraically, the value(s) of x.

Understand the Problem

The question is asking to find the value(s) of x algebraically given the dimensions of a rectangle where the area is stated to be 6 square centimeters. The length and width of the rectangle are expressed in terms of x.

Answer

The values of $ x $ are $ x = -5 $ and $ x = 2 $.

Steps to Solve

Identify dimensions of the rectangle

The dimensions of the rectangle are given in terms of $x$.

Length = $x + 4$
Width = $x - 1$

Set up the area equation

The area ( A ) of a rectangle is calculated using the formula: $$ A = \text{Length} \times \text{Width} $$

So, we have: $$ 6 = (x + 4)(x - 1) $$

Expand the equation

Now we will expand the right side of the equation: $$ 6 = x^2 - x + 4x - 4 $$ This simplifies to: $$ 6 = x^2 + 3x - 4 $$

Rearrange to form a quadratic equation

To set the equation to zero, subtract 6 from both sides: $$ 0 = x^2 + 3x - 10 $$

Factor the quadratic equation

Now we will factor the quadratic equation: $$ 0 = (x + 5)(x - 2) $$

Solve for ( x )

Set each factor to zero:

( x + 5 = 0 ) ⇒ ( x = -5 )
( x - 2 = 0 ) ⇒ ( x = 2 )

The values of ( x ) are ( x = -5 ) and ( x = 2 ).

More Information

Finding the area of a rectangle is a common application of algebra. In this case, we expressed the dimensions in terms of ( x ) and solved for ( x ) to find possible values that give the specified area of 6 square centimeters.

Tips

Forgetting to set up the equation properly with the correct length and width can lead to incorrect results.
Not checking if the values of ( x ) are valid dimensions (e.g., negative lengths).

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