Answer :
To determine how many integers between [tex]\(-10\)[/tex] and [tex]\(10\)[/tex] inclusive meet the stipulated conditions, let's proceed step-by-step:
### Step 1: Solve the quadratic equation
We start by solving the quadratic equation [tex]\(x^2 + 7x + 10 = 0\)[/tex].
To factorize the quadratic equation:
[tex]\[ x^2 + 7x + 10 = (x + 2)(x + 5) = 0 \][/tex]
Setting each factor equal to zero, we get:
[tex]\[ x + 2 = 0 \quad \text{or} \quad x + 5 = 0 \][/tex]
[tex]\[ x = -2 \quad \text{or} \quad x = -5 \][/tex]
Thus, the solutions to the quadratic equation [tex]\(x^2 + 7x + 10 = 0\)[/tex] are [tex]\(x = -2\)[/tex] and [tex]\(x = -5\)[/tex].
### Step 2: Check the absolute value of each solution
Next, we check the absolute values of these solutions:
[tex]\[ |-2| = 2 \][/tex]
[tex]\[ |-5| = 5 \][/tex]
### Step 3: Determine if the absolute values are composite numbers
A composite number is a positive integer greater than 1 that has positive divisors other than 1 and itself.
Now we identify if the absolute values are composite:
- [tex]\(2\)[/tex] is not composite because its only divisors are [tex]\(1\)[/tex] and [tex]\(2\)[/tex] (it is prime).
- [tex]\(5\)[/tex] is also not composite because its only divisors are [tex]\(1\)[/tex] and [tex]\(5\)[/tex] (it is prime).
### Step 4: Analyze the range and criteria
We need integers between [tex]\(-10\)[/tex] and [tex]\(10\)[/tex] that:
1. Are solutions to the equation [tex]\(x^2 + 7x + 10 = 0\)[/tex].
2. Have an absolute value that is a composite number.
Since none of the solutions [tex]\(x = -2\)[/tex] and [tex]\(x = -5\)[/tex] have absolute values that are composite numbers, they do not meet both conditions.
### Conclusion
Therefore, the number of integers between [tex]\(-10\)[/tex] and [tex]\(10\)[/tex] that satisfy both conditions is:
[tex]\[ \boxed{0} \][/tex]
### Step 1: Solve the quadratic equation
We start by solving the quadratic equation [tex]\(x^2 + 7x + 10 = 0\)[/tex].
To factorize the quadratic equation:
[tex]\[ x^2 + 7x + 10 = (x + 2)(x + 5) = 0 \][/tex]
Setting each factor equal to zero, we get:
[tex]\[ x + 2 = 0 \quad \text{or} \quad x + 5 = 0 \][/tex]
[tex]\[ x = -2 \quad \text{or} \quad x = -5 \][/tex]
Thus, the solutions to the quadratic equation [tex]\(x^2 + 7x + 10 = 0\)[/tex] are [tex]\(x = -2\)[/tex] and [tex]\(x = -5\)[/tex].
### Step 2: Check the absolute value of each solution
Next, we check the absolute values of these solutions:
[tex]\[ |-2| = 2 \][/tex]
[tex]\[ |-5| = 5 \][/tex]
### Step 3: Determine if the absolute values are composite numbers
A composite number is a positive integer greater than 1 that has positive divisors other than 1 and itself.
Now we identify if the absolute values are composite:
- [tex]\(2\)[/tex] is not composite because its only divisors are [tex]\(1\)[/tex] and [tex]\(2\)[/tex] (it is prime).
- [tex]\(5\)[/tex] is also not composite because its only divisors are [tex]\(1\)[/tex] and [tex]\(5\)[/tex] (it is prime).
### Step 4: Analyze the range and criteria
We need integers between [tex]\(-10\)[/tex] and [tex]\(10\)[/tex] that:
1. Are solutions to the equation [tex]\(x^2 + 7x + 10 = 0\)[/tex].
2. Have an absolute value that is a composite number.
Since none of the solutions [tex]\(x = -2\)[/tex] and [tex]\(x = -5\)[/tex] have absolute values that are composite numbers, they do not meet both conditions.
### Conclusion
Therefore, the number of integers between [tex]\(-10\)[/tex] and [tex]\(10\)[/tex] that satisfy both conditions is:
[tex]\[ \boxed{0} \][/tex]
