Question
Which of the following is a solution to the quadratic equation 2x2−5x=6
? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses
x=−1.03
x equals negative 1.03
x=−0.89
x equals negative 0.89
x=0.89
x equals 0.89
x=1.03
? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses
x=−1.03
x equals negative 1.03
x=−0.89
x equals negative 0.89
x=0.89
x equals 0.89
x=1.03
Answers
Answer
How many real solutions does the following quadratic equation have?
4x2+x+3=0
(1 point)
Responses
two real solutions
two real solutions
no real solutions
no real solutions
three real solutions
three real solutions
one real solution
4x2+x+3=0
(1 point)
Responses
two real solutions
two real solutions
no real solutions
no real solutions
three real solutions
three real solutions
one real solution
Answer
Given the quadratic equation y=5x2+8x+9
, solve for the y-coordinate of the parabola when x=−3
.(1 point)
(−3
,
)
, solve for the y-coordinate of the parabola when x=−3
.(1 point)
(−3
,
)
Answer
Given this table of values, complete the table for the quadratic equation y=3x2−6x+9
to find the values of the high or low point of the parabola.
x y
−3
54
−2
33
−1
0
1
2 9
(1 point)
The high or low point of this parabola is (
,
).
to find the values of the high or low point of the parabola.
x y
−3
54
−2
33
−1
0
1
2 9
(1 point)
The high or low point of this parabola is (
,
).
Answer
Use the image to answer the question.
A parabola is graphed on a coordinate plane. The x-axis ranges from negative 25 to 10 in increments of 5. The y-axis ranges from negative 10 to 60 in increments of 10.
Identify the vertex and axis of symmetry of the parabola.
(1 point)
axis of symmetry: x=
vertex: (
)
A parabola is graphed on a coordinate plane. The x-axis ranges from negative 25 to 10 in increments of 5. The y-axis ranges from negative 10 to 60 in increments of 10.
Identify the vertex and axis of symmetry of the parabola.
(1 point)
axis of symmetry: x=
vertex: (
)
Answer
Which of the following is a point you can generate to create the parabola for the quadratic equation y=−6x2+3x+8
?(1 point)
Responses
(−3,−55)
left parenthesis negative 3 comma negative 55 right parentheis
(−3,323)
left parenthesis negative 3 comma 323 right parentheis
(−6,206)
left parenthesis negative 6 comma 206 right parentheis
(−6,8)
?(1 point)
Responses
(−3,−55)
left parenthesis negative 3 comma negative 55 right parentheis
(−3,323)
left parenthesis negative 3 comma 323 right parentheis
(−6,206)
left parenthesis negative 6 comma 206 right parentheis
(−6,8)
Answered by
GPT-4o mini
To solve the quadratic equation \( 2x^2 - 5x - 6 = 0 \), we can either factor it, complete the square, or use the quadratic formula. Here, we will use the quadratic formula, which is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( 2x^2 - 5x - 6 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = -5 \)
- \( c = -6 \)
Now, we can calculate the discriminant (\( b^2 - 4ac \)):
\[
b^2 - 4ac = (-5)^2 - 4(2)(-6) = 25 + 48 = 73
\]
Next, substitute the values into the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{73}}{2(2)} = \frac{5 \pm \sqrt{73}}{4}
\]
Calculating \( \sqrt{73} \):
\[
\sqrt{73} \approx 8.544
\]
Now, we can find the two potential solutions:
1. \( x = \frac{5 + 8.544}{4} = \frac{13.544}{4} \approx 3.386 \)
2. \( x = \frac{5 - 8.544}{4} = \frac{-3.544}{4} \approx -0.886 \)
Rounding to the nearest hundredth, we get:
1. \( x \approx 3.39 \)
2. \( x \approx -0.89 \)
So, the approximate solutions are \( x \approx 3.39 \) and \( x \approx -0.89 \).
From the given options, the one that matches is:
**x = -0.89.**
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the equation \( 2x^2 - 5x - 6 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = -5 \)
- \( c = -6 \)
Now, we can calculate the discriminant (\( b^2 - 4ac \)):
\[
b^2 - 4ac = (-5)^2 - 4(2)(-6) = 25 + 48 = 73
\]
Next, substitute the values into the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{73}}{2(2)} = \frac{5 \pm \sqrt{73}}{4}
\]
Calculating \( \sqrt{73} \):
\[
\sqrt{73} \approx 8.544
\]
Now, we can find the two potential solutions:
1. \( x = \frac{5 + 8.544}{4} = \frac{13.544}{4} \approx 3.386 \)
2. \( x = \frac{5 - 8.544}{4} = \frac{-3.544}{4} \approx -0.886 \)
Rounding to the nearest hundredth, we get:
1. \( x \approx 3.39 \)
2. \( x \approx -0.89 \)
So, the approximate solutions are \( x \approx 3.39 \) and \( x \approx -0.89 \).
From the given options, the one that matches is:
**x = -0.89.**
Answered by
GPT-4o mini
To determine the number of real solutions to the quadratic equation \(4x^2 + x + 3 = 0\), we can use the discriminant method. The discriminant (\(D\)) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula:
\[
D = b^2 - 4ac
\]
For the equation \(4x^2 + x + 3 = 0\), we have:
- \(a = 4\)
- \(b = 1\)
- \(c = 3\)
Now calculate the discriminant:
\[
D = (1)^2 - 4(4)(3) = 1 - 48 = -47
\]
Since the discriminant is negative (\(D < 0\)), this indicates that there are **no real solutions** to the equation.
Therefore, the correct response is:
**no real solutions.**
\[
D = b^2 - 4ac
\]
For the equation \(4x^2 + x + 3 = 0\), we have:
- \(a = 4\)
- \(b = 1\)
- \(c = 3\)
Now calculate the discriminant:
\[
D = (1)^2 - 4(4)(3) = 1 - 48 = -47
\]
Since the discriminant is negative (\(D < 0\)), this indicates that there are **no real solutions** to the equation.
Therefore, the correct response is:
**no real solutions.**
Answered by
GPT-4o mini
To find the y-coordinate of the parabola represented by the quadratic equation \( y = 5x^2 + 8x + 9 \) when \( x = -3 \), we can substitute \(-3\) into the equation for \(x\):
\[
y = 5(-3)^2 + 8(-3) + 9
\]
Calculating step by step:
1. Calculate \((-3)^2\):
\[
(-3)^2 = 9
\]
2. Multiply by \(5\):
\[
5 \cdot 9 = 45
\]
3. Calculate \(8 \cdot (-3)\):
\[
8 \cdot (-3) = -24
\]
4. Now substitute these values back into the equation:
\[
y = 45 - 24 + 9
\]
5. Perform the addition:
\[
45 - 24 = 21
\]
\[
21 + 9 = 30
\]
Thus, when \( x = -3 \), the y-coordinate is \( 30 \).
The coordinates of the point on the parabola are:
\[
(-3, 30)
\]
\[
y = 5(-3)^2 + 8(-3) + 9
\]
Calculating step by step:
1. Calculate \((-3)^2\):
\[
(-3)^2 = 9
\]
2. Multiply by \(5\):
\[
5 \cdot 9 = 45
\]
3. Calculate \(8 \cdot (-3)\):
\[
8 \cdot (-3) = -24
\]
4. Now substitute these values back into the equation:
\[
y = 45 - 24 + 9
\]
5. Perform the addition:
\[
45 - 24 = 21
\]
\[
21 + 9 = 30
\]
Thus, when \( x = -3 \), the y-coordinate is \( 30 \).
The coordinates of the point on the parabola are:
\[
(-3, 30)
\]
Answered by
GPT-4o mini
To find the high or low point of the parabola given by the quadratic equation \(y = 3x^2 - 6x + 9\), we can calculate its vertex. The vertex of a parabola in the form \(y = ax^2 + bx + c\) can be found using the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
For the equation \(y = 3x^2 - 6x + 9\):
- \(a = 3\)
- \(b = -6\)
Substituting these values into the formula gives us:
\[
x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1
\]
Next, we will substitute \(x = 1\) back into the equation to find the corresponding \(y\)-coordinate:
\[
y = 3(1)^2 - 6(1) + 9
\]
\[
y = 3(1) - 6 + 9 = 3 - 6 + 9 = 6
\]
Thus, the vertex, which represents the low point of the parabola (since \(a = 3 > 0\), the parabola opens upwards), is:
\[
(1, 6)
\]
Now, let's complete the table for the quadratic equation using the given values:
### Completing the Table:
- For \(x = -3\):
\[
y = 3(-3)^2 - 6(-3) + 9 = 3(9) + 18 + 9 = 27 + 18 + 9 = 54
\]
- For \(x = -2\):
\[
y = 3(-2)^2 - 6(-2) + 9 = 3(4) + 12 + 9 = 12 + 12 + 9 = 33
\]
- For \(x = -1\):
\[
y = 3(-1)^2 - 6(-1) + 9 = 3(1) + 6 + 9 = 3 + 6 + 9 = 18
\]
- For \(x = 0\):
\[
y = 3(0)^2 - 6(0) + 9 = 0 + 0 + 9 = 9
\]
- For \(x = 1\):
\[
y = 3(1)^2 - 6(1) + 9 = 3(1) - 6 + 9 = 3 - 6 + 9 = 6
\]
- For \(x = 2\):
\[
y = 3(2)^2 - 6(2) + 9 = 3(4) - 12 + 9 = 12 - 12 + 9 = 9
\]
### Final Table:
| x | y |
|-----|-----|
| -3 | 54 |
| -2 | 33 |
| -1 | 18 |
| 0 | 9 |
| 1 | 6 |
| 2 | 9 |
### Conclusion:
The high or low point of this parabola is \( (1, 6) \).
\[
x = -\frac{b}{2a}
\]
For the equation \(y = 3x^2 - 6x + 9\):
- \(a = 3\)
- \(b = -6\)
Substituting these values into the formula gives us:
\[
x = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1
\]
Next, we will substitute \(x = 1\) back into the equation to find the corresponding \(y\)-coordinate:
\[
y = 3(1)^2 - 6(1) + 9
\]
\[
y = 3(1) - 6 + 9 = 3 - 6 + 9 = 6
\]
Thus, the vertex, which represents the low point of the parabola (since \(a = 3 > 0\), the parabola opens upwards), is:
\[
(1, 6)
\]
Now, let's complete the table for the quadratic equation using the given values:
### Completing the Table:
- For \(x = -3\):
\[
y = 3(-3)^2 - 6(-3) + 9 = 3(9) + 18 + 9 = 27 + 18 + 9 = 54
\]
- For \(x = -2\):
\[
y = 3(-2)^2 - 6(-2) + 9 = 3(4) + 12 + 9 = 12 + 12 + 9 = 33
\]
- For \(x = -1\):
\[
y = 3(-1)^2 - 6(-1) + 9 = 3(1) + 6 + 9 = 3 + 6 + 9 = 18
\]
- For \(x = 0\):
\[
y = 3(0)^2 - 6(0) + 9 = 0 + 0 + 9 = 9
\]
- For \(x = 1\):
\[
y = 3(1)^2 - 6(1) + 9 = 3(1) - 6 + 9 = 3 - 6 + 9 = 6
\]
- For \(x = 2\):
\[
y = 3(2)^2 - 6(2) + 9 = 3(4) - 12 + 9 = 12 - 12 + 9 = 9
\]
### Final Table:
| x | y |
|-----|-----|
| -3 | 54 |
| -2 | 33 |
| -1 | 18 |
| 0 | 9 |
| 1 | 6 |
| 2 | 9 |
### Conclusion:
The high or low point of this parabola is \( (1, 6) \).
Answered by
GPT-4o mini
To determine the vertex and axis of symmetry of a parabola graphed on a coordinate plane, we typically look for:
1. **Vertex**: The point at which the parabola changes direction, which is either a high point (maximum) or a low point (minimum).
2. **Axis of Symmetry**: A vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. The equation of the axis of symmetry can be represented as \(x = k\), where \(k\) is the x-coordinate of the vertex.
### Step-by-Step Approach:
1. **Find the Vertex**:
- Look for the highest or lowest point on the parabola (the peak or trough). The coordinates of that point give you the vertex.
2. **Determine the Axis of Symmetry**:
- The x-coordinate of the vertex is the same for the axis of symmetry. Therefore, if the vertex is \((h, k)\), then the axis of symmetry is \(x = h\).
### Example (Hypothetical):
Suppose you found from the graph that:
- The vertex is at the point \((-5, 20)\).
- This means the vertex is \((-5, 20)\) and the axis of symmetry would be the line \(x = -5\).
### Answer Format:
If you can provide the specific coordinates of the vertex you observe in the image (if you have access to it), I can help you finalize the response with the correct values. However, a typical answer would look like this based on hypothetical coordinates:
- **Axis of symmetry**: \(x = -5\)
- **Vertex**: \((-5, 20)\)
If you provide the vertex coordinates from the graph, I can help you to finalize the response accordingly!
1. **Vertex**: The point at which the parabola changes direction, which is either a high point (maximum) or a low point (minimum).
2. **Axis of Symmetry**: A vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. The equation of the axis of symmetry can be represented as \(x = k\), where \(k\) is the x-coordinate of the vertex.
### Step-by-Step Approach:
1. **Find the Vertex**:
- Look for the highest or lowest point on the parabola (the peak or trough). The coordinates of that point give you the vertex.
2. **Determine the Axis of Symmetry**:
- The x-coordinate of the vertex is the same for the axis of symmetry. Therefore, if the vertex is \((h, k)\), then the axis of symmetry is \(x = h\).
### Example (Hypothetical):
Suppose you found from the graph that:
- The vertex is at the point \((-5, 20)\).
- This means the vertex is \((-5, 20)\) and the axis of symmetry would be the line \(x = -5\).
### Answer Format:
If you can provide the specific coordinates of the vertex you observe in the image (if you have access to it), I can help you finalize the response with the correct values. However, a typical answer would look like this based on hypothetical coordinates:
- **Axis of symmetry**: \(x = -5\)
- **Vertex**: \((-5, 20)\)
If you provide the vertex coordinates from the graph, I can help you to finalize the response accordingly!
Answered by
GPT-4o mini
To determine which of the given points lies on the parabola defined by the equation \( y = -6x^2 + 3x + 8 \), we can substitute the x-coordinates of each point into the equation and see if the resulting y-coordinate matches.
### Given equation:
\[
y = -6x^2 + 3x + 8
\]
### Check each point:
1. **For the point \((-3, -55)\)**:
\[
y = -6(-3)^2 + 3(-3) + 8
\]
\[
= -6(9) - 9 + 8
\]
\[
= -54 - 9 + 8
\]
\[
= -55 \quad \text{(matches)}
\]
2. **For the point \((-3, 323)\)**:
(Already checked **\(-3\)** above, gives **\(-55\)**, so this does not match)
3. **For the point \((-6, 206)\)**:
\[
y = -6(-6)^2 + 3(-6) + 8
\]
\[
= -6(36) - 18 + 8
\]
\[
= -216 - 18 + 8
\]
\[
= -226 \quad \text{(does not match)}
\]
4. **For the point \((-6, 8)\)**:
\[
y = -6(-6)^2 + 3(-6) + 8
\]
(This calculation is the same as above, which gives \(-226\), so here also does not match.)
### Conclusion:
The point that satisfies the equation \( y = -6x^2 + 3x + 8 \) is:
**\((-3, -55)\)**.
So, the correct response is:
\((-3, -55)\).
### Given equation:
\[
y = -6x^2 + 3x + 8
\]
### Check each point:
1. **For the point \((-3, -55)\)**:
\[
y = -6(-3)^2 + 3(-3) + 8
\]
\[
= -6(9) - 9 + 8
\]
\[
= -54 - 9 + 8
\]
\[
= -55 \quad \text{(matches)}
\]
2. **For the point \((-3, 323)\)**:
(Already checked **\(-3\)** above, gives **\(-55\)**, so this does not match)
3. **For the point \((-6, 206)\)**:
\[
y = -6(-6)^2 + 3(-6) + 8
\]
\[
= -6(36) - 18 + 8
\]
\[
= -216 - 18 + 8
\]
\[
= -226 \quad \text{(does not match)}
\]
4. **For the point \((-6, 8)\)**:
\[
y = -6(-6)^2 + 3(-6) + 8
\]
(This calculation is the same as above, which gives \(-226\), so here also does not match.)
### Conclusion:
The point that satisfies the equation \( y = -6x^2 + 3x + 8 \) is:
**\((-3, -55)\)**.
So, the correct response is:
\((-3, -55)\).
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