Question 3 A:
To determine whether \( x = -3 \) is a solution to the quadratic equation \( -3x^2 - 9x = 0 \), we can substitute \( x = -3 \) into the equation and check if the left-hand side equals the right-hand side.
Substituting \( x = -3 \):
\[ -3(-3)^2 - 9(-3) = -3(9) + 27 = -27 + 27 = 0 \]
Since both sides of the equation equal 0, the correct reasoning is:
Yes, \( x = -3 \) is a solution because substituting it back into the equation results in the following: \[ -3(-3)^2 - 9(-3) = 0 \quad \Rightarrow \quad 0 = 0 \quad \text{(True)} \]
Question 4 A:
To find the number of solutions for the equation \( 3x^2 = 21 \), we can divide both sides by 3:
\[ x^2 = 7 \]
Next, we can take the square root of both sides:
\[ x = \sqrt{7} \quad \text{and} \quad x = -\sqrt{7} \]
This means there are two solutions to the equation. Therefore, the correct response is:
Two.
Question 5 A:
To find how many solutions are in the solution set for the equation \( 2(x-5)^2 + 2 = 20 \), we can first subtract 2 from both sides:
\[ 2(x - 5)^2 = 18 \]
Next, divide by 2:
\[ (x - 5)^2 = 9 \]
Now take the square root of both sides:
\[ x - 5 = 3 \quad \Rightarrow \quad x = 8 \] \[ x - 5 = -3 \quad \Rightarrow \quad x = 2 \]
Thus, there are two solutions to this equation. Therefore, the correct response is:
Two.
Summary of Answers:
- Question 3 A: Yes, \( x = -3 \) is a solution.
- Question 4 A: Two.
- Question 5 A: Two.