Let's solve this step by step, finding the correct values and expressions to complete the sentences.
Step 1: The equation representing the area of the cookie sheet is already given.
[tex]\[ x^2 + 4x = 192 \][/tex]
Step 2: Factor the left-hand side of the equation to match the given area (192 square inches).
[tex]\[ x(x + 4) = 192 \][/tex]
So the equation in this step becomes:
[tex]\[ x(x + 4) = 192 \][/tex]
Step 3: Rewrite the equation in the form of a standard quadratic equation:
[tex]\[ x^2 + 4x - 192 = 0 \][/tex]
Step 4: Factor the quadratic equation. We are looking for two numbers that multiply to -192 and add up to 4.
The two numbers are 16 and -12, because [tex]\(16 \times -12 = -192\)[/tex] and [tex]\(16 - 12 = 4\)[/tex].
So we can factor the quadratic as:
[tex]\[ (x + 16)(x - 12) = 0 \][/tex]
Final Steps: Solve for [tex]\(x\)[/tex] to determine the lengths of the sides:
1. [tex]\( x + 16 = 0 \)[/tex]
[tex]\[ x = -16 \][/tex] (This cannot be a dimension since a dimension cannot be negative)
2. [tex]\( x - 12 = 0 \)[/tex]
[tex]\[ x = 12 \][/tex]
So the length is 12 inches.
Since the length is 12 inches, and we know the expression [tex]\(x + 4\)[/tex] represents the width:
[tex]\[ \text{If } x = 12 \text{, then width } = 12 + 4 = 16 \text{ inches} \][/tex]
Thus:
Because the length can't be [tex]\(-16\)[/tex], the length is [tex]\(12\)[/tex] inches, and the width is [tex]\(16\)[/tex] inches.
Now, let's place the values and expressions into the final sentences:
Step 1: [tex]\( x^2 + 4x = 192 \)[/tex]
Step 2: [tex]\( x ( x + 4 ) = 192 \)[/tex]
The length is [tex]\( x \)[/tex].
The width is [tex]\( x + 4 \)[/tex].
Step 3: [tex]\( x^2 + 4x - 192 = 0 \)[/tex]
Step 4: [tex]\( ( x + 16 )( x - 12 ) = 0 \)[/tex]
Because the length can't be [tex]\(-16\)[/tex], the length is [tex]\( 12 \)[/tex], and the width is [tex]\( 16 \)[/tex].
