To determine which statements about the quadratic equation [tex]\(x^2 + 15x + 50 = 0\)[/tex] are correct, let's analyze each statement step by step:
### Statement 1: [tex]\(x = -5\)[/tex]
To verify if [tex]\(x = -5\)[/tex] is a solution to the equation, we substitute [tex]\(x = -5\)[/tex] into the equation:
[tex]\[
(-5)^2 + 15(-5) + 50 = 25 - 75 + 50 = 0
\][/tex]
This calculation shows that the left side of the equation equals zero when [tex]\(x = -5\)[/tex]. Therefore, [tex]\(x = -5\)[/tex] is a correct solution.
### Statement 2: [tex]\(x - 11 = 0\)[/tex]
This implies that [tex]\(x = 11\)[/tex]. Let's check if [tex]\(x = 11\)[/tex] is a solution to the equation by substituting [tex]\(x = 11\)[/tex] into the equation:
[tex]\[
11^2 + 15 \cdot 11 + 50 = 121 + 165 + 50
\][/tex]
Calculating the above expression:
[tex]\[
121 + 165 + 50 = 336 \neq 0
\][/tex]
Since [tex]\(336 \neq 0\)[/tex], [tex]\(x = 11\)[/tex] is not a solution. Therefore, this statement is incorrect.
### Statement 3: [tex]\(bc = 10\)[/tex]
We know the quadratic equation in its standard form is [tex]\(ax^2 + bx + c = 0\)[/tex]. Given the coefficients [tex]\(a = 1\)[/tex], [tex]\(b = 15\)[/tex], and [tex]\(c = 50\)[/tex], let's check the product [tex]\(b \cdot c\)[/tex]:
[tex]\[
b \cdot c = 15 \cdot 50 = 750
\][/tex]
Since [tex]\(750 \neq 10\)[/tex], the statement [tex]\(bc = 10\)[/tex] is incorrect.
### Statement 4: [tex]\(x - 10 = 0\)[/tex]
This implies that [tex]\(x = 10\)[/tex]. Let's check if [tex]\(x = 10\)[/tex] is a solution to the equation by substituting [tex]\(x = 10\)[/tex] into the equation:
[tex]\[
10^2 + 15 \cdot 10 + 50 = 100 + 150 + 50
\][/tex]
Calculating the above expression:
[tex]\[
100 + 150 + 50 = 300 \neq 0
\][/tex]
Since [tex]\(300 \neq 0\)[/tex], [tex]\(x = 10\)[/tex] is not a solution. Therefore, this statement is incorrect.
### Conclusion
After evaluating each statement, we conclude:
1. [tex]\(x = -5\)[/tex] is correct.
2. [tex]\(x - 11 = 0\)[/tex] is incorrect.
3. [tex]\(b \cdot c = 10\)[/tex] is incorrect.
4. [tex]\(x - 10 = 0\)[/tex] is incorrect.
So the correct statement is:
[tex]\[
\text{1. } x = -5
\][/tex]
