Question
Question
Each of the following quadratic equations can be solved by factoring. Which equation has only one distinct solution?(1 point)
Responses
x2−4=0
x squared minus 4 equals 0
x2−5x+4=0
x squared minus 5 x plus 4 equals 0
x2+5x+4=0
x squared plus 5 x plus 4 equals 0
x2+4x+4=0
Each of the following quadratic equations can be solved by factoring. Which equation has only one distinct solution?(1 point)
Responses
x2−4=0
x squared minus 4 equals 0
x2−5x+4=0
x squared minus 5 x plus 4 equals 0
x2+5x+4=0
x squared plus 5 x plus 4 equals 0
x2+4x+4=0
Answers
Answer
Question
Solve the following quadratic equation using the perfect square trinomial pattern: 14x2+2x+4=0
.(1 point)
x=
Solve the following quadratic equation using the perfect square trinomial pattern: 14x2+2x+4=0
.(1 point)
x=
Answer
Gwen was asked to solve 3x2−20=7
by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point)
Responses
x=−3
, x=3
x equals negative 3 , x equals 3
x=3
x equals 3
x=−3
x equals negative 3
x=9
, x=−9
by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point)
Responses
x=−3
, x=3
x equals negative 3 , x equals 3
x=3
x equals 3
x=−3
x equals negative 3
x=9
, x=−9
Answer
Complete the square to identify the two values that solve the following quadratic equation: x2−16x−13=4
.(1 point)
x=
.(1 point)
x=
Answer
Which of the following is a solution to the quadratic equation 4x2+2x=10
? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses
x=−1.85
x equals negative 1.85
x=1.04
x equals 1.04
x=−0.77
x equals negative 0.77
x=1.15
? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses
x=−1.85
x equals negative 1.85
x=1.04
x equals 1.04
x=−0.77
x equals negative 0.77
x=1.15
Answer
What are the key points on the graph of y=x2−16x+28
? Name the vertex, x-intercepts, and y-intercept.(2 points)
vertex: (
)
(smaller) x-intercept: (
)
(larger) x-intercept: (
)
y-intercept: (
)
? Name the vertex, x-intercepts, and y-intercept.(2 points)
vertex: (
)
(smaller) x-intercept: (
)
(larger) x-intercept: (
)
y-intercept: (
)
Answered by
GPT-4o mini
To determine which quadratic equation has only one distinct solution, we look for equations that can be factored into a perfect square form, which would indicate that the solutions are repeated (i.e., the same root).
1. **Equation: \( x^2 - 4 = 0 \)**
This factors to \( (x - 2)(x + 2) = 0 \)
Solutions: \( x = 2 \) and \( x = -2 \) (two distinct solutions)
2. **Equation: \( x^2 - 5x + 4 = 0 \)**
This factors to \( (x - 1)(x - 4) = 0 \)
Solutions: \( x = 1 \) and \( x = 4 \) (two distinct solutions)
3. **Equation: \( x^2 + 5x + 4 = 0 \)**
This factors to \( (x + 1)(x + 4) = 0 \)
Solutions: \( x = -1 \) and \( x = -4 \) (two distinct solutions)
4. **Equation: \( x^2 + 4x + 4 = 0 \)**
This factors to \( (x + 2)(x + 2) = 0 \)
Solutions: \( x = -2 \) (one distinct solution, repeated root)
Since only the last equation, \( x^2 + 4x + 4 = 0 \), yields only one distinct solution, the answer is:
**\( x^2 + 4x + 4 = 0 \)**
1. **Equation: \( x^2 - 4 = 0 \)**
This factors to \( (x - 2)(x + 2) = 0 \)
Solutions: \( x = 2 \) and \( x = -2 \) (two distinct solutions)
2. **Equation: \( x^2 - 5x + 4 = 0 \)**
This factors to \( (x - 1)(x - 4) = 0 \)
Solutions: \( x = 1 \) and \( x = 4 \) (two distinct solutions)
3. **Equation: \( x^2 + 5x + 4 = 0 \)**
This factors to \( (x + 1)(x + 4) = 0 \)
Solutions: \( x = -1 \) and \( x = -4 \) (two distinct solutions)
4. **Equation: \( x^2 + 4x + 4 = 0 \)**
This factors to \( (x + 2)(x + 2) = 0 \)
Solutions: \( x = -2 \) (one distinct solution, repeated root)
Since only the last equation, \( x^2 + 4x + 4 = 0 \), yields only one distinct solution, the answer is:
**\( x^2 + 4x + 4 = 0 \)**
Answered by
GPT-4o mini
To solve the quadratic equation \(14x^2 + 2x + 4 = 0\) using the perfect square trinomial pattern, we can start by rearranging and simplifying the equation.
1. First, divide the entire equation by 2 to make the numbers simpler:
\[
7x^2 + x + 2 = 0
\]
2. Rearrange the equation:
\[
7x^2 + x = -2
\]
3. To complete the square, we first divide all terms involving \(x\) by \(7\):
\[
x^2 + \frac{1}{7}x = -\frac{2}{7}
\]
4. Now, we'll complete the square. The coefficient of \(x\) is \(\frac{1}{7}\). To complete the square, take half of that coefficient, square it, and add it to both sides:
- Half of \(\frac{1}{7}\) is \(\frac{1}{14}\).
- Squaring \(\frac{1}{14}\) gives \(\frac{1}{196}\).
So, we add \(\frac{1}{196}\) to both sides:
\[
x^2 + \frac{1}{7}x + \frac{1}{196} = -\frac{2}{7} + \frac{1}{196}
\]
5. We need to find a common denominator to add the fractions on the right side. The common denominator will be \(196\):
- Convert \(-\frac{2}{7}\) to \(-\frac{56}{196}\).
- So, we have:
\[
-\frac{2}{7} + \frac{1}{196} = -\frac{56}{196} + \frac{1}{196} = -\frac{55}{196}
\]
6. Now our equation is:
\[
x^2 + \frac{1}{7}x + \frac{1}{196} = -\frac{55}{196}
\]
7. The left-hand side can be factored:
\[
\left(x + \frac{1}{14}\right)^2 = -\frac{55}{196}
\]
Since the right side is negative, there are no real solutions to this equation.
The solutions are complex. We can express the square root of the negative number:
\[
\left(x + \frac{1}{14}\right)^2 = \frac{55}{196} i
\]
Now take the square root of both sides:
\[
x + \frac{1}{14} = \pm \sqrt{\frac{55}{196}} i
\]
\[
x + \frac{1}{14} = \pm \frac{\sqrt{55}}{14} i
\]
8. Finally, solving for \(x\):
\[
x = -\frac{1}{14} \pm \frac{\sqrt{55}}{14} i
\]
Thus, the solutions to the quadratic equation \(14x^2 + 2x + 4 = 0\) are:
\[
x = -\frac{1}{14} + \frac{\sqrt{55}}{14} i \quad \text{and} \quad x = -\frac{1}{14} - \frac{\sqrt{55}}{14} i
\]
1. First, divide the entire equation by 2 to make the numbers simpler:
\[
7x^2 + x + 2 = 0
\]
2. Rearrange the equation:
\[
7x^2 + x = -2
\]
3. To complete the square, we first divide all terms involving \(x\) by \(7\):
\[
x^2 + \frac{1}{7}x = -\frac{2}{7}
\]
4. Now, we'll complete the square. The coefficient of \(x\) is \(\frac{1}{7}\). To complete the square, take half of that coefficient, square it, and add it to both sides:
- Half of \(\frac{1}{7}\) is \(\frac{1}{14}\).
- Squaring \(\frac{1}{14}\) gives \(\frac{1}{196}\).
So, we add \(\frac{1}{196}\) to both sides:
\[
x^2 + \frac{1}{7}x + \frac{1}{196} = -\frac{2}{7} + \frac{1}{196}
\]
5. We need to find a common denominator to add the fractions on the right side. The common denominator will be \(196\):
- Convert \(-\frac{2}{7}\) to \(-\frac{56}{196}\).
- So, we have:
\[
-\frac{2}{7} + \frac{1}{196} = -\frac{56}{196} + \frac{1}{196} = -\frac{55}{196}
\]
6. Now our equation is:
\[
x^2 + \frac{1}{7}x + \frac{1}{196} = -\frac{55}{196}
\]
7. The left-hand side can be factored:
\[
\left(x + \frac{1}{14}\right)^2 = -\frac{55}{196}
\]
Since the right side is negative, there are no real solutions to this equation.
The solutions are complex. We can express the square root of the negative number:
\[
\left(x + \frac{1}{14}\right)^2 = \frac{55}{196} i
\]
Now take the square root of both sides:
\[
x + \frac{1}{14} = \pm \sqrt{\frac{55}{196}} i
\]
\[
x + \frac{1}{14} = \pm \frac{\sqrt{55}}{14} i
\]
8. Finally, solving for \(x\):
\[
x = -\frac{1}{14} \pm \frac{\sqrt{55}}{14} i
\]
Thus, the solutions to the quadratic equation \(14x^2 + 2x + 4 = 0\) are:
\[
x = -\frac{1}{14} + \frac{\sqrt{55}}{14} i \quad \text{and} \quad x = -\frac{1}{14} - \frac{\sqrt{55}}{14} i
\]
Answered by
GPT-4o mini
To solve the equation \(3x^2 - 20 = 7\) by factoring using the difference of squares pattern, we can follow these steps:
1. **Rearrange the equation**:
Start by moving 7 to the left side:
\[
3x^2 - 20 - 7 = 0
\]
Simplifying this, we get:
\[
3x^2 - 27 = 0
\]
2. **Factor out the common term**:
Factor out \(3\):
\[
3(x^2 - 9) = 0
\]
3. **Recognize the difference of squares**:
Notice that \(x^2 - 9\) can be factored further using the difference of squares:
\[
3(x - 3)(x + 3) = 0
\]
4. **Set each factor to zero**:
Set the factors equal to zero:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
Thus, the solutions to the equation \(3x^2 - 20 = 7\) are:
- \(x = 3\)
- \(x = -3\)
**Therefore, the correct response is:**
**\(x = -3, x = 3\)**
1. **Rearrange the equation**:
Start by moving 7 to the left side:
\[
3x^2 - 20 - 7 = 0
\]
Simplifying this, we get:
\[
3x^2 - 27 = 0
\]
2. **Factor out the common term**:
Factor out \(3\):
\[
3(x^2 - 9) = 0
\]
3. **Recognize the difference of squares**:
Notice that \(x^2 - 9\) can be factored further using the difference of squares:
\[
3(x - 3)(x + 3) = 0
\]
4. **Set each factor to zero**:
Set the factors equal to zero:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
Thus, the solutions to the equation \(3x^2 - 20 = 7\) are:
- \(x = 3\)
- \(x = -3\)
**Therefore, the correct response is:**
**\(x = -3, x = 3\)**
Answered by
GPT-4o mini
To solve the quadratic equation \(x^2 - 16x - 13 = 4\) by completing the square, follow these steps:
### Step 1: Rearrange the equation
Move \(4\) to the left side of the equation:
\[
x^2 - 16x - 13 - 4 = 0
\]
This simplifies to:
\[
x^2 - 16x - 17 = 0
\]
### Step 2: Isolate the quadratic and linear terms
We'll work with the equation in this form:
\[
x^2 - 16x = 17
\]
### Step 3: Complete the square
To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides:
- The coefficient of \(x\) is \(-16\).
- Half of \(-16\) is \(-8\).
- Squaring \(-8\) gives \(64\).
Add \(64\) to both sides:
\[
x^2 - 16x + 64 = 17 + 64
\]
This simplifies to:
\[
x^2 - 16x + 64 = 81
\]
### Step 4: Factor the left side
Now the left side can be factored as a perfect square:
\[
(x - 8)^2 = 81
\]
### Step 5: Take the square root of both sides
Taking the square root of both sides gives us:
\[
x - 8 = \pm 9
\]
### Step 6: Solve for \(x\)
Now we solve for \(x\) by setting up two equations:
1. \(x - 8 = 9\)
\[
x = 9 + 8 = 17
\]
2. \(x - 8 = -9\)
\[
x = -9 + 8 = -1
\]
### Final Solutions
Thus, the two solutions to the equation are:
\[
x = 17 \quad \text{and} \quad x = -1
\]
**The solutions are \(x = 17\) and \(x = -1\).**
### Step 1: Rearrange the equation
Move \(4\) to the left side of the equation:
\[
x^2 - 16x - 13 - 4 = 0
\]
This simplifies to:
\[
x^2 - 16x - 17 = 0
\]
### Step 2: Isolate the quadratic and linear terms
We'll work with the equation in this form:
\[
x^2 - 16x = 17
\]
### Step 3: Complete the square
To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides:
- The coefficient of \(x\) is \(-16\).
- Half of \(-16\) is \(-8\).
- Squaring \(-8\) gives \(64\).
Add \(64\) to both sides:
\[
x^2 - 16x + 64 = 17 + 64
\]
This simplifies to:
\[
x^2 - 16x + 64 = 81
\]
### Step 4: Factor the left side
Now the left side can be factored as a perfect square:
\[
(x - 8)^2 = 81
\]
### Step 5: Take the square root of both sides
Taking the square root of both sides gives us:
\[
x - 8 = \pm 9
\]
### Step 6: Solve for \(x\)
Now we solve for \(x\) by setting up two equations:
1. \(x - 8 = 9\)
\[
x = 9 + 8 = 17
\]
2. \(x - 8 = -9\)
\[
x = -9 + 8 = -1
\]
### Final Solutions
Thus, the two solutions to the equation are:
\[
x = 17 \quad \text{and} \quad x = -1
\]
**The solutions are \(x = 17\) and \(x = -1\).**
Answered by
GPT-4o mini
To find the solution to the quadratic equation \(4x^2 + 2x = 10\), we start by rearranging the equation into the standard form \(ax^2 + bx + c = 0\):
1. **Subtract 10 from both sides**:
\[
4x^2 + 2x - 10 = 0
\]
Next, we can either factor or use the quadratic formula to solve for \(x\). Since this equation does not easily factor, we'll use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 4\), \(b = 2\), and \(c = -10\).
### Step 1: Calculate the discriminant
\[
b^2 - 4ac = (2)^2 - 4(4)(-10) = 4 + 160 = 164
\]
### Step 2: Apply the quadratic formula
Now plug in the values:
\[
x = \frac{-2 \pm \sqrt{164}}{2 \times 4} = \frac{-2 \pm \sqrt{164}}{8}
\]
### Step 3: Simplify \(\sqrt{164}\)
We can simplify \(\sqrt{164}\):
\[
\sqrt{164} = \sqrt{4 \cdot 41} = 2\sqrt{41}
\]
Thus, we have:
\[
x = \frac{-2 \pm 2\sqrt{41}}{8}
\]
This simplifies to:
\[
x = \frac{-1 \pm \sqrt{41}}{4}
\]
### Step 4: Calculate the two possible values for \(x\)
Now we will find both possible solutions:
1. **First solution**:
\[
x = \frac{-1 + \sqrt{41}}{4}
\]
2. **Second solution**:
\[
x = \frac{-1 - \sqrt{41}}{4}
\]
Now calculating these values:
1. **Calculate \(\sqrt{41} \approx 6.4\)**:
- \(x_1 = \frac{-1 + 6.4}{4} = \frac{5.4}{4} \approx 1.35\)
- \(x_2 = \frac{-1 - 6.4}{4} = \frac{-7.4}{4} \approx -1.85\)
### Step 5: Round the values as necessary
- The approximate values are \(x_1 \approx 1.35\) and \(x_2 \approx -1.85\).
### Final Step: Compare with the given options
Now comparing with the available options:
- \(x = -1.85\)
- \(x = 1.04\)
- \(x = -0.77\)
- \(x = 1.15\)
The option that is closest to a calculated value is:
\[
\boxed{x = -1.85}
\]
Thus, **the correct answer is \(x = -1.85\)**.
1. **Subtract 10 from both sides**:
\[
4x^2 + 2x - 10 = 0
\]
Next, we can either factor or use the quadratic formula to solve for \(x\). Since this equation does not easily factor, we'll use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 4\), \(b = 2\), and \(c = -10\).
### Step 1: Calculate the discriminant
\[
b^2 - 4ac = (2)^2 - 4(4)(-10) = 4 + 160 = 164
\]
### Step 2: Apply the quadratic formula
Now plug in the values:
\[
x = \frac{-2 \pm \sqrt{164}}{2 \times 4} = \frac{-2 \pm \sqrt{164}}{8}
\]
### Step 3: Simplify \(\sqrt{164}\)
We can simplify \(\sqrt{164}\):
\[
\sqrt{164} = \sqrt{4 \cdot 41} = 2\sqrt{41}
\]
Thus, we have:
\[
x = \frac{-2 \pm 2\sqrt{41}}{8}
\]
This simplifies to:
\[
x = \frac{-1 \pm \sqrt{41}}{4}
\]
### Step 4: Calculate the two possible values for \(x\)
Now we will find both possible solutions:
1. **First solution**:
\[
x = \frac{-1 + \sqrt{41}}{4}
\]
2. **Second solution**:
\[
x = \frac{-1 - \sqrt{41}}{4}
\]
Now calculating these values:
1. **Calculate \(\sqrt{41} \approx 6.4\)**:
- \(x_1 = \frac{-1 + 6.4}{4} = \frac{5.4}{4} \approx 1.35\)
- \(x_2 = \frac{-1 - 6.4}{4} = \frac{-7.4}{4} \approx -1.85\)
### Step 5: Round the values as necessary
- The approximate values are \(x_1 \approx 1.35\) and \(x_2 \approx -1.85\).
### Final Step: Compare with the given options
Now comparing with the available options:
- \(x = -1.85\)
- \(x = 1.04\)
- \(x = -0.77\)
- \(x = 1.15\)
The option that is closest to a calculated value is:
\[
\boxed{x = -1.85}
\]
Thus, **the correct answer is \(x = -1.85\)**.
Answered by
GPT-4o mini
To find the key points on the graph of the quadratic function \(y = x^2 - 16x + 28\), we will determine the vertex, x-intercepts, and y-intercept.
### Step 1: Find the Vertex
The vertex of a quadratic function in the form \(y = ax^2 + bx + c\) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Here, \(a = 1\) and \(b = -16\).
Calculating the x-coordinate of the vertex:
\[
x = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8
\]
Next, we substitute \(x = 8\) back into the function to find the y-coordinate:
\[
y = (8)^2 - 16(8) + 28 = 64 - 128 + 28 = -36
\]
Thus, the vertex is:
\[
\text{Vertex: } (8, -36)
\]
### Step 2: Find the x-intercepts
To find the x-intercepts, set \(y = 0\):
\[
0 = x^2 - 16x + 28
\]
We can solve this quadratic using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
In our case, \(a = 1\), \(b = -16\), and \(c = 28\).
Calculating the discriminant:
\[
b^2 - 4ac = (-16)^2 - 4(1)(28) = 256 - 112 = 144
\]
Now applying the quadratic formula:
\[
x = \frac{-(-16) \pm \sqrt{144}}{2(1)} = \frac{16 \pm 12}{2}
\]
Calculating the two solutions:
1. \(x = \frac{16 + 12}{2} = \frac{28}{2} = 14\)
2. \(x = \frac{16 - 12}{2} = \frac{4}{2} = 2\)
Thus, the x-intercepts are:
\[
\text{Smaller x-intercept: } (2, 0)
\]
\[
\text{Larger x-intercept: } (14, 0)
\]
### Step 3: Find the y-intercept
To find the y-intercept, set \(x = 0\):
\[
y = (0)^2 - 16(0) + 28 = 28
\]
Thus, the y-intercept is:
\[
\text{Y-intercept: } (0, 28)
\]
### Summary of Key Points
- **Vertex**: \( (8, -36) \)
- **Smaller x-intercept**: \( (2, 0) \)
- **Larger x-intercept**: \( (14, 0) \)
- **Y-intercept**: \( (0, 28) \)
So the final answer is:
- Vertex: \( (8, -36) \)
- Smaller x-intercept: \( (2, 0) \)
- Larger x-intercept: \( (14, 0) \)
- Y-intercept: \( (0, 28) \)
### Step 1: Find the Vertex
The vertex of a quadratic function in the form \(y = ax^2 + bx + c\) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Here, \(a = 1\) and \(b = -16\).
Calculating the x-coordinate of the vertex:
\[
x = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8
\]
Next, we substitute \(x = 8\) back into the function to find the y-coordinate:
\[
y = (8)^2 - 16(8) + 28 = 64 - 128 + 28 = -36
\]
Thus, the vertex is:
\[
\text{Vertex: } (8, -36)
\]
### Step 2: Find the x-intercepts
To find the x-intercepts, set \(y = 0\):
\[
0 = x^2 - 16x + 28
\]
We can solve this quadratic using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
In our case, \(a = 1\), \(b = -16\), and \(c = 28\).
Calculating the discriminant:
\[
b^2 - 4ac = (-16)^2 - 4(1)(28) = 256 - 112 = 144
\]
Now applying the quadratic formula:
\[
x = \frac{-(-16) \pm \sqrt{144}}{2(1)} = \frac{16 \pm 12}{2}
\]
Calculating the two solutions:
1. \(x = \frac{16 + 12}{2} = \frac{28}{2} = 14\)
2. \(x = \frac{16 - 12}{2} = \frac{4}{2} = 2\)
Thus, the x-intercepts are:
\[
\text{Smaller x-intercept: } (2, 0)
\]
\[
\text{Larger x-intercept: } (14, 0)
\]
### Step 3: Find the y-intercept
To find the y-intercept, set \(x = 0\):
\[
y = (0)^2 - 16(0) + 28 = 28
\]
Thus, the y-intercept is:
\[
\text{Y-intercept: } (0, 28)
\]
### Summary of Key Points
- **Vertex**: \( (8, -36) \)
- **Smaller x-intercept**: \( (2, 0) \)
- **Larger x-intercept**: \( (14, 0) \)
- **Y-intercept**: \( (0, 28) \)
So the final answer is:
- Vertex: \( (8, -36) \)
- Smaller x-intercept: \( (2, 0) \)
- Larger x-intercept: \( (14, 0) \)
- Y-intercept: \( (0, 28) \)
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.