Certainly! Let's break down the problem step-by-step.
1. Jessica and Martha together have 30 cookies in total.
2. Let's assume Jessica started with [tex]\( x \)[/tex] cookies.
3. Therefore, Martha started with [tex]\( 30 - x \)[/tex] cookies.
4. Both of them ate 6 cookies each.
After eating 6 cookies:
- Jessica has [tex]\( x - 6 \)[/tex] cookies left.
- Martha has [tex]\( (30 - x - 6) = 24 - x \)[/tex] cookies left.
The problem states that the product of the number of cookies left in each bag is not more than 80. This can be written as:
[tex]\[ (x - 6)(24 - x) \leq 80 \][/tex]
Expanding and simplifying this product:
[tex]\[ (x - 6)(24 - x) \leq 80 \][/tex]
[tex]\[ 24x - x^2 - 144 + 6x \leq 80\ ]
Combining like terms:
\[ -x^2 + 30x - 144 \leq 80 \][/tex]
Rearranging to standard form:
[tex]\[ -x^2 + 30x - 224 \leq 0 \][/tex]
This changes to:
[tex]\[ x^2 - 30x + 224 \geq 0 \][/tex]
To solve this inequality, we need to set it to zero and solve for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 30x + 224 = 0 \][/tex]
The solutions for [tex]\( x \)[/tex] from solving the quadratic equation [tex]\( x^2 - 30x + 224 = 0 \)[/tex] are the roots of the polynomial. Let’s consider these solutions to understand the boundary conditions.
From the given solution from the Python calculations, the complete form of the quadratic equation is:
[tex]\[ x^2 - 30x + 224 \geq 0 \][/tex]
Now, understanding from the result:
- The coefficient completing the square is 30.
- The difference in the number of cookies Jessica has more than Martha after they both ate 6 cookies is at least 2.
Therefore, the final inequality and the number of additional cookies Jessica has:
1. The inequality that describes the relationship between the number of cookies each one of them has is:
[tex]\[ x^2 - 30x + 224 \geq 0 \][/tex]
2. Jessica has at least:
[tex]\[ 2 \][/tex]
cookies more than Martha.
