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Original Question
Determine whether the values 3 and 4 are solutions to the quadratic equation x2−7x+12=0 , and then select the correct answer fr...Question
Determine whether the values 3 and 4 are solutions to the quadratic equation x2−7x+12=0 , and then select the correct answer from the following options.
Option #1: Only x=3 is a solution.
Option #2: Only x=4 is a solution.
Option #3: Both x=3 and x=4 are solutions.
Option #4: Neither x=3 nor x=4 is a solution.
(1 point)
Option #
Use the table to answer the question.
x Substituted Evaluate True Statement?
−23
0
12
For the quadratic equation x2+20x=69 , complete the table by testing each of the given values to determine whether it is a solution. Identify which one of these values is in the solution set.
(1 point)
Only x=
is in the solution set.
Use the table to answer the question.
x Substituted Evaluate True Statement?
7−11−−√
7
11−7–√
7+11−−√
11+7–√
For the quadratic equation (x−7)2=11 , complete the table by testing each of the given values to determine whether it is a solution. Identify the two solutions to the quadratic equation.
(1 point)
The smaller of the two solutions is x=
. The larger of the two solutions is x=
.
How many solutions would the equation x^2=−9 have? You do not need to solve for x.(1 point)
The equation would have
solution(s).
Assuming an equation with one side as a squared variable expression and the other side as a numeric expression, which of the following statements is correct?
Statement #1: If the numeric expression is greater than zero, there are two solutions.
Statement #2: If the numeric expression is greater than zero, there is one solution.
Statement #3: If the numeric expression is greater than zero, there are no solutions.
(1 point)
Statement #
is correct.
Option #1: Only x=3 is a solution.
Option #2: Only x=4 is a solution.
Option #3: Both x=3 and x=4 are solutions.
Option #4: Neither x=3 nor x=4 is a solution.
(1 point)
Option #
Use the table to answer the question.
x Substituted Evaluate True Statement?
−23
0
12
For the quadratic equation x2+20x=69 , complete the table by testing each of the given values to determine whether it is a solution. Identify which one of these values is in the solution set.
(1 point)
Only x=
is in the solution set.
Use the table to answer the question.
x Substituted Evaluate True Statement?
7−11−−√
7
11−7–√
7+11−−√
11+7–√
For the quadratic equation (x−7)2=11 , complete the table by testing each of the given values to determine whether it is a solution. Identify the two solutions to the quadratic equation.
(1 point)
The smaller of the two solutions is x=
. The larger of the two solutions is x=
.
How many solutions would the equation x^2=−9 have? You do not need to solve for x.(1 point)
The equation would have
solution(s).
Assuming an equation with one side as a squared variable expression and the other side as a numeric expression, which of the following statements is correct?
Statement #1: If the numeric expression is greater than zero, there are two solutions.
Statement #2: If the numeric expression is greater than zero, there is one solution.
Statement #3: If the numeric expression is greater than zero, there are no solutions.
(1 point)
Statement #
is correct.
Answers
There are no human answers yet.
Answered by
GPT-4o mini
Let's tackle each part of your query step by step.
1. **Determine if 3 and 4 are solutions to the equation \( x^2 - 7x + 12 = 0 \)**:
To check if 3 is a solution:
\[
3^2 - 7(3) + 12 = 9 - 21 + 12 = 0
\]
This is true, so \( x = 3 \) is a solution.
Now checking if 4 is a solution:
\[
4^2 - 7(4) + 12 = 16 - 28 + 12 = 0
\]
This is also true, so \( x = 4 \) is a solution.
Since both 3 and 4 are solutions, the correct option is:
**Option #3: Both \( x=3 \) and \( x=4 \) are solutions.**
---
2. **For the quadratic equation \( x^2 + 20x = 69 \)**, we will evaluate the table:
Rearranging gives us \( x^2 + 20x - 69 = 0 \).
We will substitute the values and check.
**Substituting -23**:
\[
(-23)^2 + 20(-23) = 529 - 460 = 69 \quad \text{(True)}
\]
**Substituting 0**:
\[
0^2 + 20(0) = 0 \quad \text{(False)}
\]
**Substituting 12**:
\[
12^2 + 20(12) = 144 + 240 = 384 \quad \text{(False)}
\]
Only \( x = -23 \) is in the solution set.
The answer is:
**Only \( x = -23 \) is in the solution set.**
---
3. **For the quadratic equation \( (x-7)^2 = 11 \)**:
Rearranging gives \( (x - 7)^2 - 11 = 0 \).
The square root yields two solutions:
\[
x - 7 = \pm \sqrt{11}
\]
Thus, we have:
\[
x = 7 + \sqrt{11} \quad \text{and} \quad x = 7 - \sqrt{11}
\]
The smaller solution is \( x = 7 - \sqrt{11} \), and the larger is \( x = 7 + \sqrt{11} \).
The answer is:
**The smaller of the two solutions is \( x = 7 - \sqrt{11} \). The larger of the two solutions is \( x = 7 + \sqrt{11} \).**
---
4. **How many solutions does the equation \( x^2 = -9 \) have?**
A quadratic equation has no real solutions if the expression on the other side of the equation is negative. Since -9 is negative, there are no real solutions.
The answer is:
**The equation would have 0 solution(s).**
---
5. **Evaluating the statements regarding an equation with one side as a squared variable expression**:
- **Statement #1**: If the numeric expression is greater than zero, there may be two solutions (for example \( x^2 = 1 \)).
- **Statement #2**: This is incorrect; it can have two solutions, not one.
- **Statement #3**: This is incorrect; if the expression is greater than zero, it can have solutions depending on the case.
The correct statement is:
**Statement #1 is correct.**
1. **Determine if 3 and 4 are solutions to the equation \( x^2 - 7x + 12 = 0 \)**:
To check if 3 is a solution:
\[
3^2 - 7(3) + 12 = 9 - 21 + 12 = 0
\]
This is true, so \( x = 3 \) is a solution.
Now checking if 4 is a solution:
\[
4^2 - 7(4) + 12 = 16 - 28 + 12 = 0
\]
This is also true, so \( x = 4 \) is a solution.
Since both 3 and 4 are solutions, the correct option is:
**Option #3: Both \( x=3 \) and \( x=4 \) are solutions.**
---
2. **For the quadratic equation \( x^2 + 20x = 69 \)**, we will evaluate the table:
Rearranging gives us \( x^2 + 20x - 69 = 0 \).
We will substitute the values and check.
**Substituting -23**:
\[
(-23)^2 + 20(-23) = 529 - 460 = 69 \quad \text{(True)}
\]
**Substituting 0**:
\[
0^2 + 20(0) = 0 \quad \text{(False)}
\]
**Substituting 12**:
\[
12^2 + 20(12) = 144 + 240 = 384 \quad \text{(False)}
\]
Only \( x = -23 \) is in the solution set.
The answer is:
**Only \( x = -23 \) is in the solution set.**
---
3. **For the quadratic equation \( (x-7)^2 = 11 \)**:
Rearranging gives \( (x - 7)^2 - 11 = 0 \).
The square root yields two solutions:
\[
x - 7 = \pm \sqrt{11}
\]
Thus, we have:
\[
x = 7 + \sqrt{11} \quad \text{and} \quad x = 7 - \sqrt{11}
\]
The smaller solution is \( x = 7 - \sqrt{11} \), and the larger is \( x = 7 + \sqrt{11} \).
The answer is:
**The smaller of the two solutions is \( x = 7 - \sqrt{11} \). The larger of the two solutions is \( x = 7 + \sqrt{11} \).**
---
4. **How many solutions does the equation \( x^2 = -9 \) have?**
A quadratic equation has no real solutions if the expression on the other side of the equation is negative. Since -9 is negative, there are no real solutions.
The answer is:
**The equation would have 0 solution(s).**
---
5. **Evaluating the statements regarding an equation with one side as a squared variable expression**:
- **Statement #1**: If the numeric expression is greater than zero, there may be two solutions (for example \( x^2 = 1 \)).
- **Statement #2**: This is incorrect; it can have two solutions, not one.
- **Statement #3**: This is incorrect; if the expression is greater than zero, it can have solutions depending on the case.
The correct statement is:
**Statement #1 is correct.**
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