Question
For the quadratic equation xΒ² - x = 12, which of the following statements is correct?
Statement #1: The equation has one solution.
Statement #2: The equation has two solutions.
Statement #3: The equation has no solutions.
(1 point)
Responses
Statement #1
Statement #1
Statement #2
Statement #2
Statement #3
Statement #1: The equation has one solution.
Statement #2: The equation has two solutions.
Statement #3: The equation has no solutions.
(1 point)
Responses
Statement #1
Statement #1
Statement #2
Statement #2
Statement #3
Answers
GPT-4o mini
To determine how many solutions the quadratic equation \( x^2 - x - 12 = 0 \) has, we can first rewrite the equation in standard form and then calculate the discriminant.
1. Rearranging the equation:
\[
x^2 - x - 12 = 0
\]
2. The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is calculated as:
\[
D = b^2 - 4ac
\]
where \( a = 1 \), \( b = -1 \), and \( c = -12 \).
3. Substituting in the values:
\[
D = (-1)^2 - 4(1)(-12) = 1 + 48 = 49
\]
The discriminant \( D = 49 \) is positive, which indicates that the quadratic equation has two distinct real solutions.
Therefore, the correct statement is:
**Statement #2: The equation has two solutions.**
1. Rearranging the equation:
\[
x^2 - x - 12 = 0
\]
2. The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is calculated as:
\[
D = b^2 - 4ac
\]
where \( a = 1 \), \( b = -1 \), and \( c = -12 \).
3. Substituting in the values:
\[
D = (-1)^2 - 4(1)(-12) = 1 + 48 = 49
\]
The discriminant \( D = 49 \) is positive, which indicates that the quadratic equation has two distinct real solutions.
Therefore, the correct statement is:
**Statement #2: The equation has two solutions.**
Solve x2
π₯
2
= 78
7
8
by inspection. What are the two real solutions?(1 point)
Responses
{β74, 74
β
7
4
,
7
4
}
{β74, 74
β
7
4
,
7
4
}
{β72, 72
β
7
2
,
7
2
}
{β72, 72
β
7
2
,
7
2
}
{β78, 78
β
7
8
,
7
8
}
{β78, 78
β
7
8
,
7
8
}
{β74, 7β4
β
7
4
,
7
β
4
}
π₯
2
= 78
7
8
by inspection. What are the two real solutions?(1 point)
Responses
{β74, 74
β
7
4
,
7
4
}
{β74, 74
β
7
4
,
7
4
}
{β72, 72
β
7
2
,
7
2
}
{β72, 72
β
7
2
,
7
2
}
{β78, 78
β
7
8
,
7
8
}
{β78, 78
β
7
8
,
7
8
}
{β74, 7β4
β
7
4
,
7
β
4
}
GPT-4o mini
To solve the equation \( x^2 = 78 \) by inspection, we are looking for two real numbers whose squares equal 78.
We can start by calculating the square root of 78:
\[
x = \pm \sqrt{78}
\]
Now, let's break down \(\sqrt{78}\):
\[
\sqrt{78} = \sqrt{39 \cdot 2} \approx 8.83 \quad (\text{since } 8^2 = 64 \text{ and } 9^2 = 81)
\]
Thus, the two real solutions are:
\[
x = \sqrt{78} \quad \text{and} \quad x = -\sqrt{78}
\]
Now letβs evaluate the provided options to see which one corresponds to \( \pm\sqrt{78} \):
1. **{β74, 74}**
2. **{β72, 72}**
3. **{β78, 78}**
4. **{β74, 7β4}**
None of these clearly match \( \pm\sqrt{78} \), which is approximately \( \pm 8.83 \). It seems like the options do not include the correct solutions.
Therefore, based on solving \( x^2 = 78 \), the two real solutions are:
\[
x = -\sqrt{78} \quad \text{and} \quad x = \sqrt{78}
\]
If the options are incorrect, itβs important to note that the exact values should be \( \{ -\sqrt{78}, \sqrt{78} \} \). If you meant to refer to approximate values, it would look more like \( \{ -8.83, 8.83 \} \).
As such, none of the given responses accurately represent the solutions to \( x^2 = 78 \).
We can start by calculating the square root of 78:
\[
x = \pm \sqrt{78}
\]
Now, let's break down \(\sqrt{78}\):
\[
\sqrt{78} = \sqrt{39 \cdot 2} \approx 8.83 \quad (\text{since } 8^2 = 64 \text{ and } 9^2 = 81)
\]
Thus, the two real solutions are:
\[
x = \sqrt{78} \quad \text{and} \quad x = -\sqrt{78}
\]
Now letβs evaluate the provided options to see which one corresponds to \( \pm\sqrt{78} \):
1. **{β74, 74}**
2. **{β72, 72}**
3. **{β78, 78}**
4. **{β74, 7β4}**
None of these clearly match \( \pm\sqrt{78} \), which is approximately \( \pm 8.83 \). It seems like the options do not include the correct solutions.
Therefore, based on solving \( x^2 = 78 \), the two real solutions are:
\[
x = -\sqrt{78} \quad \text{and} \quad x = \sqrt{78}
\]
If the options are incorrect, itβs important to note that the exact values should be \( \{ -\sqrt{78}, \sqrt{78} \} \). If you meant to refer to approximate values, it would look more like \( \{ -8.83, 8.83 \} \).
As such, none of the given responses accurately represent the solutions to \( x^2 = 78 \).