To determine how many integers between 10 and 80 (inclusive) meet both conditions, we can follow these steps:
1. Solve the Equation [tex]\( x^3 + x^2 + 11x = 0 \)[/tex]:
Start by factoring the equation:
[tex]\[ x^3 + x^2 + 11x = 0 \][/tex]
First, factor out the common [tex]\( x \)[/tex]:
[tex]\[ x(x^2 + x + 11) = 0 \][/tex]
This yields the solutions:
[tex]\[ x = 0 \quad \text{or} \quad x^2 + x + 11 = 0 \][/tex]
Next, solve the quadratic equation [tex]\( x^2 + x + 11 = 0 \)[/tex].
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For our quadratic equation [tex]\( x^2 + x + 11 \)[/tex]:
[tex]\[ a = 1, \quad b = 1, \quad c = 11 \][/tex]
Calculate the discriminant:
[tex]\[ b^2 - 4ac = 1^2 - 4(1)(11) = 1 - 44 = -43 \][/tex]
Since the discriminant is negative, [tex]\( x^2 + x + 11 = 0 \)[/tex] has no real solutions. Therefore, the only real solution to the original equation is:
[tex]\[ x = 0 \][/tex]
Now, we need to find integers between 10 and 80 that have a digit equal to 0 and for which the sum of the digits is at least 4.
2. List Relevant Integers That Have a Digit of 0:
Consider integers between 10 and 80 that have a digit 0:
[tex]\[ 10, 20, 30, 40, 50, 60, 70 \][/tex]
3. Check If Their Digit Sum is At Least 4:
- [tex]\(10\)[/tex]: [tex]\(1 + 0 = 1\)[/tex] (does not satisfy the condition)
- [tex]\(20\)[/tex]: [tex]\(2 + 0 = 2\)[/tex] (does not satisfy the condition)
- [tex]\(30\)[/tex]: [tex]\(3 + 0 = 3\)[/tex] (does not satisfy the condition)
- [tex]\(40\)[/tex]: [tex]\(4 + 0 = 4\)[/tex] (satisfies the condition)
- [tex]\(50\)[/tex]: [tex]\(5 + 0 = 5\)[/tex] (satisfies the condition)
- [tex]\(60\)[/tex]: [tex]\(6 + 0 = 6\)[/tex] (satisfies the condition)
- [tex]\(70\)[/tex]: [tex]\(7 + 0 = 7\)[/tex] (satisfies the condition)
The integers that meet both conditions are: 40, 50, 60, 70
4. Count the Relevant Integers:
There are 4 integers that meet both conditions:
[tex]\[ 40, 50, 60, 70 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{4} \][/tex]
However, none of the provided options is matching our solution. There might be a mistake in transcription or missing options in the question provided. Revising or verifying the given choices with the calculated result is advised.
