Question


Yes, x=−3 is a solution because substituting it back into the equation results in the following:

x Substituted Evaluate True Statement?
−3 −3(−3)2−9(−3)=0 0=0 True

true or false
Answer
Show how many solutions are in the solution set for the equation 3x^2=21 .(1 point)
Answer
Show how many solutions are in the solution set for 2(x−5)^2+2=20 .(1 point)
Responses
Answered by GPT-4o mini
The statement is true.

To verify:

1. Substitute \( x = -3 \) into the equation \( -3x^2 - 9x = 0 \).
2. Calculate:
\[
-3(-3)^2 - 9(-3) \\
= -3(9) + 27 \\
= -27 + 27 \\
= 0
\]

The left side evaluates to \( 0 \), which equals the right side (also \( 0 \)). Therefore, the statement confirms that \( x = -3 \) is indeed a solution to the equation.
Answered by GPT-4o mini
To determine the number of solutions for the equation \( 3x^2 = 21 \), we can follow these steps:

1. **Simplify the equation**:
\[
3x^2 = 21
\]
Divide both sides by \( 3 \):
\[
x^2 = 7
\]

2. **Solve for \( x \)** by taking the square root of both sides:
\[
x = \pm\sqrt{7}
\]

This results in two solutions:
\[
x = \sqrt{7} \quad \text{and} \quad x = -\sqrt{7}
\]

Therefore, the solution set contains **2 solutions**.
Answered by GPT-4o mini
To determine the number of solutions for the equation \( 2(x-5)^2 + 2 = 20 \), we'll follow these steps:

1. **Subtract 2 from both sides**:
\[
2(x-5)^2 + 2 - 2 = 20 - 2
\]
This simplifies to:
\[
2(x-5)^2 = 18
\]

2. **Divide both sides by 2**:
\[
(x-5)^2 = 9
\]

3. **Take the square root of both sides**:
\[
x-5 = \pm 3
\]

4. **Solve for \( x \)**:
- For \( x - 5 = 3 \):
\[
x = 3 + 5 = 8
\]
- For \( x - 5 = -3 \):
\[
x = -3 + 5 = 2
\]

So the solutions are \( x = 8 \) and \( x = 2 \).

Therefore, the solution set contains **2 solutions**.