Asked by x
Complete the square to solve the following quadratic equation: x2−6x−20=20.(1 point)
Responses
x=52, x=−46
x equals 52 , x equals negative 46
x=10, x=−4
x equals 10 , x equals negative 4
x=3+210−−√, 3−210−−√
x equals 3 plus 2 Start Root 10 End Root comma 3 minus 2 Start Root 10 End Root
x=−10, x=4
Responses
x=52, x=−46
x equals 52 , x equals negative 46
x=10, x=−4
x equals 10 , x equals negative 4
x=3+210−−√, 3−210−−√
x equals 3 plus 2 Start Root 10 End Root comma 3 minus 2 Start Root 10 End Root
x=−10, x=4
Answers
Answered by
x
Using the quadratic formula, find the solution to 3x^2+18x+24=0.(1 point)
Responses
x=4, x=2
x equals 4 , x equals 2
x=−4
x equals negative 4
x=−2, x=−4
x equals negative 2 , x equals negative 4
x=−2
Responses
x=4, x=2
x equals 4 , x equals 2
x=−4
x equals negative 4
x=−2, x=−4
x equals negative 2 , x equals negative 4
x=−2
Answered by
x
Which of the following is a solution to the quadratic equation −x^2+x=−10? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses
x=0.77
x equals 0.77
x=0.27
x equals 0.27
x=3.7
x equals 3.7
x=1.7
Responses
x=0.77
x equals 0.77
x=0.27
x equals 0.27
x=3.7
x equals 3.7
x=1.7
Answered by
x
Which of the following coordinate pairs is a point on the parabola y=−4x^2−53x−56?(1 point)
Responses
(−4,−56)
left parenthesis negative 4 comma negative 56 right parenthesis
(−4,220)
left parenthesis negative 4 comma 220 right parenthesis
(−1,13)
left parenthesis neagtive 1 comma 13 right parenthesis
(−1,−7)
left parenthesis negative1 comma negative 7 right parenthesis
Responses
(−4,−56)
left parenthesis negative 4 comma negative 56 right parenthesis
(−4,220)
left parenthesis negative 4 comma 220 right parenthesis
(−1,13)
left parenthesis neagtive 1 comma 13 right parenthesis
(−1,−7)
left parenthesis negative1 comma negative 7 right parenthesis
Answered by
x
Determine the zero(s) of the quadratic function y=−5x^2+16x.(1 point)
Responses
(0,5)(0,16)
left parenthesis 0 comma 5 right parenthesis left parenthesis 0 comma 16 right parenthesis
(0,0)(3.2,0)
left parenthesis 0 comma 0 right parenthesis left parenthesis 3.2 comma 0 right parenthesis
(−3.2,0)
left parenthesis negative 3.2 comma 0 right parenthesis
There are no zeros.
Responses
(0,5)(0,16)
left parenthesis 0 comma 5 right parenthesis left parenthesis 0 comma 16 right parenthesis
(0,0)(3.2,0)
left parenthesis 0 comma 0 right parenthesis left parenthesis 3.2 comma 0 right parenthesis
(−3.2,0)
left parenthesis negative 3.2 comma 0 right parenthesis
There are no zeros.
Answered by
x
Identify the vertex of y=4(x−25)^2−61 .(1 point)
Responses
(−100,−61)
left parenthesis negative 100 comma negative 61 right parenthesis
(4,−25)
left parenthesis 4 comma negative 25 right parenthesis
(25,−61)
left parenthesis 25 comma negative 61 right parenthesis
(−25,−61)
Responses
(−100,−61)
left parenthesis negative 100 comma negative 61 right parenthesis
(4,−25)
left parenthesis 4 comma negative 25 right parenthesis
(25,−61)
left parenthesis 25 comma negative 61 right parenthesis
(−25,−61)
Answered by
GPT-4o mini
To solve the equation \( x^2 - 6x - 20 = 20 \) by completing the square, first, let's rewrite the equation in standard form:
1. Move 20 to the left side:
\[
x^2 - 6x - 20 - 20 = 0 \quad \Rightarrow \quad x^2 - 6x - 40 = 0
\]
2. Now, we will complete the square for \( x^2 - 6x \):
- Take half of the coefficient of \( x \) (which is -6), square it, and add it.
- Half of -6 is -3, and squaring it gives us 9.
3. Rewrite the quadratic:
\[
x^2 - 6x + 9 - 9 - 40 = 0 \quad \Rightarrow \quad (x - 3)^2 - 49 = 0
\]
4. Set the completed square equal to 49:
\[
(x - 3)^2 = 49
\]
5. Take the square root of both sides:
\[
x - 3 = 7 \quad \text{or} \quad x - 3 = -7
\]
6. Solve for \( x \):
- \( x - 3 = 7 \) gives \( x = 10 \)
- \( x - 3 = -7 \) gives \( x = -4 \)
So the solutions to the quadratic equation are:
\[
x = 10 \quad \text{and} \quad x = -4
\]
Therefore, the correct response is:
**x=10, x=−4**
**x equals 10 , x equals negative 4**
1. Move 20 to the left side:
\[
x^2 - 6x - 20 - 20 = 0 \quad \Rightarrow \quad x^2 - 6x - 40 = 0
\]
2. Now, we will complete the square for \( x^2 - 6x \):
- Take half of the coefficient of \( x \) (which is -6), square it, and add it.
- Half of -6 is -3, and squaring it gives us 9.
3. Rewrite the quadratic:
\[
x^2 - 6x + 9 - 9 - 40 = 0 \quad \Rightarrow \quad (x - 3)^2 - 49 = 0
\]
4. Set the completed square equal to 49:
\[
(x - 3)^2 = 49
\]
5. Take the square root of both sides:
\[
x - 3 = 7 \quad \text{or} \quad x - 3 = -7
\]
6. Solve for \( x \):
- \( x - 3 = 7 \) gives \( x = 10 \)
- \( x - 3 = -7 \) gives \( x = -4 \)
So the solutions to the quadratic equation are:
\[
x = 10 \quad \text{and} \quad x = -4
\]
Therefore, the correct response is:
**x=10, x=−4**
**x equals 10 , x equals negative 4**
Answered by
GPT-4o mini
To solve the quadratic equation \(3x^2 + 18x + 24 = 0\) using the quadratic formula, we start by identifying the coefficients \(a\), \(b\), and \(c\) from the standard form \(ax^2 + bx + c = 0\).
Here:
- \(a = 3\)
- \(b = 18\)
- \(c = 24\)
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Now, let's calculate \(b^2 - 4ac\):
\[
b^2 = 18^2 = 324
\]
\[
4ac = 4 \cdot 3 \cdot 24 = 288
\]
\[
b^2 - 4ac = 324 - 288 = 36
\]
Since the discriminant (\(b^2 - 4ac\)) is positive, we will have two real solutions. Now, we can substitute back into the quadratic formula:
\[
x = \frac{-18 \pm \sqrt{36}}{2 \cdot 3}
\]
\[
x = \frac{-18 \pm 6}{6}
\]
Now we find the two possible solutions:
1. \(x = \frac{-18 + 6}{6} = \frac{-12}{6} = -2\)
2. \(x = \frac{-18 - 6}{6} = \frac{-24}{6} = -4\)
Therefore, the solutions to the equation are:
\[
x = -2 \quad \text{and} \quad x = -4
\]
The correct response is:
**x=−2, x=−4**
**x equals negative 2 , x equals negative 4**
Here:
- \(a = 3\)
- \(b = 18\)
- \(c = 24\)
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Now, let's calculate \(b^2 - 4ac\):
\[
b^2 = 18^2 = 324
\]
\[
4ac = 4 \cdot 3 \cdot 24 = 288
\]
\[
b^2 - 4ac = 324 - 288 = 36
\]
Since the discriminant (\(b^2 - 4ac\)) is positive, we will have two real solutions. Now, we can substitute back into the quadratic formula:
\[
x = \frac{-18 \pm \sqrt{36}}{2 \cdot 3}
\]
\[
x = \frac{-18 \pm 6}{6}
\]
Now we find the two possible solutions:
1. \(x = \frac{-18 + 6}{6} = \frac{-12}{6} = -2\)
2. \(x = \frac{-18 - 6}{6} = \frac{-24}{6} = -4\)
Therefore, the solutions to the equation are:
\[
x = -2 \quad \text{and} \quad x = -4
\]
The correct response is:
**x=−2, x=−4**
**x equals negative 2 , x equals negative 4**
Answered by
GPT-4o mini
To solve the quadratic equation \(-x^2 + x = -10\), we first need to rearrange it into standard form.
1. We can add \(10\) to both sides of the equation:
\[
-x^2 + x + 10 = 0
\]
2. To make calculations easier, let's multiply the entire equation by \(-1\) to switch the signs:
\[
x^2 - x - 10 = 0
\]
Now, we can use the quadratic formula, which is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 1\), \(b = -1\), and \(c = -10\).
3. Calculate the discriminant:
\[
b^2 - 4ac = (-1)^2 - 4(1)(-10) = 1 + 40 = 41
\]
4. Substitute into the quadratic formula:
\[
x = \frac{-(-1) \pm \sqrt{41}}{2 \cdot 1}
\]
\[
x = \frac{1 \pm \sqrt{41}}{2}
\]
5. Now, calculate the two potential solutions:
- For \(x = \frac{1 + \sqrt{41}}{2}\):
\( \sqrt{41} \approx 6.403 \)
\[
x \approx \frac{1 + 6.403}{2} \approx \frac{7.403}{2} \approx 3.70
\]
- For \(x = \frac{1 - \sqrt{41}}{2}\):
\[
x \approx \frac{1 - 6.403}{2} \approx \frac{-5.403}{2} \approx -2.70
\]
Thus, rounding to the nearest hundredth, the solutions are approximately:
1. \(x \approx 3.70\)
2. \(x \approx -2.70\) (not amongst the choices)
The only solution from the provided options is:
**x=3.7**
**x equals 3.7**
1. We can add \(10\) to both sides of the equation:
\[
-x^2 + x + 10 = 0
\]
2. To make calculations easier, let's multiply the entire equation by \(-1\) to switch the signs:
\[
x^2 - x - 10 = 0
\]
Now, we can use the quadratic formula, which is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 1\), \(b = -1\), and \(c = -10\).
3. Calculate the discriminant:
\[
b^2 - 4ac = (-1)^2 - 4(1)(-10) = 1 + 40 = 41
\]
4. Substitute into the quadratic formula:
\[
x = \frac{-(-1) \pm \sqrt{41}}{2 \cdot 1}
\]
\[
x = \frac{1 \pm \sqrt{41}}{2}
\]
5. Now, calculate the two potential solutions:
- For \(x = \frac{1 + \sqrt{41}}{2}\):
\( \sqrt{41} \approx 6.403 \)
\[
x \approx \frac{1 + 6.403}{2} \approx \frac{7.403}{2} \approx 3.70
\]
- For \(x = \frac{1 - \sqrt{41}}{2}\):
\[
x \approx \frac{1 - 6.403}{2} \approx \frac{-5.403}{2} \approx -2.70
\]
Thus, rounding to the nearest hundredth, the solutions are approximately:
1. \(x \approx 3.70\)
2. \(x \approx -2.70\) (not amongst the choices)
The only solution from the provided options is:
**x=3.7**
**x equals 3.7**
Answered by
GPT-4o mini
To determine which of the given coordinate pairs is a point on the parabola represented by the equation \(y = -4x^2 - 53x - 56\), we will substitute the \(x\) values from each coordinate pair into the equation and see if the corresponding \(y\) value matches.
Let's evaluate each coordinate pair.
1. **For \((-4, -56)\)**:
\[
y = -4(-4)^2 - 53(-4) - 56
\]
\[
= -4(16) + 212 - 56
\]
\[
= -64 + 212 - 56
\]
\[
= -64 + 156 = 92 \quad (\text{not } -56)
\]
2. **For \((-4, 220)\)**:
\[
y = -4(-4)^2 - 53(-4) - 56
\]
As calculated above, \(y = 92\) (not \(220\)).
3. **For \((-1, 13)\)**:
\[
y = -4(-1)^2 - 53(-1) - 56
\]
\[
= -4(1) + 53 - 56
\]
\[
= -4 + 53 - 56
\]
\[
= -4 - 3 = -7 \quad (\text{not } 13)
\]
4. **For \((-1, -7)\)**:
Using the same calculation:
\[
y = -4(-1)^2 - 53(-1) - 56
\]
\[
= -4(1) + 53 - 56
\]
\[
= -4 + 53 - 56
\]
\[
= -4 - 3 = -7 \quad (\text{this matches } -7)
\]
Thus, the coordinate pair **\((-1, -7)\)** is a point on the parabola \(y = -4x^2 - 53x - 56\).
**The correct response is:**
**(−1,−7)**
**left parenthesis negative 1 comma negative 7 right parenthesis**
Let's evaluate each coordinate pair.
1. **For \((-4, -56)\)**:
\[
y = -4(-4)^2 - 53(-4) - 56
\]
\[
= -4(16) + 212 - 56
\]
\[
= -64 + 212 - 56
\]
\[
= -64 + 156 = 92 \quad (\text{not } -56)
\]
2. **For \((-4, 220)\)**:
\[
y = -4(-4)^2 - 53(-4) - 56
\]
As calculated above, \(y = 92\) (not \(220\)).
3. **For \((-1, 13)\)**:
\[
y = -4(-1)^2 - 53(-1) - 56
\]
\[
= -4(1) + 53 - 56
\]
\[
= -4 + 53 - 56
\]
\[
= -4 - 3 = -7 \quad (\text{not } 13)
\]
4. **For \((-1, -7)\)**:
Using the same calculation:
\[
y = -4(-1)^2 - 53(-1) - 56
\]
\[
= -4(1) + 53 - 56
\]
\[
= -4 + 53 - 56
\]
\[
= -4 - 3 = -7 \quad (\text{this matches } -7)
\]
Thus, the coordinate pair **\((-1, -7)\)** is a point on the parabola \(y = -4x^2 - 53x - 56\).
**The correct response is:**
**(−1,−7)**
**left parenthesis negative 1 comma negative 7 right parenthesis**
Answered by
GPT-4o mini
To determine the zero(s) of the quadratic function given by \( y = -5x^2 + 16x \), we need to set \( y \) to zero and solve for \( x \):
\[
-5x^2 + 16x = 0
\]
We can factor this equation:
\[
x(-5x + 16) = 0
\]
Setting each factor equal to zero gives:
1. \( x = 0 \)
2. \(-5x + 16 = 0\)
Solving for \( x \):
\[
-5x = -16 \quad \Rightarrow \quad x = \frac{16}{5} = 3.2
\]
Thus, the zeros of the function are:
\[
x = 0 \quad \text{and} \quad x = 3.2
\]
In coordinate form, these zeros can be represented as points:
\((0, 0)\) and \((3.2, 0)\).
Therefore, the correct response is:
**(0,0)(3.2,0)**
**left parenthesis 0 comma 0 right parenthesis left parenthesis 3.2 comma 0 right parenthesis**
\[
-5x^2 + 16x = 0
\]
We can factor this equation:
\[
x(-5x + 16) = 0
\]
Setting each factor equal to zero gives:
1. \( x = 0 \)
2. \(-5x + 16 = 0\)
Solving for \( x \):
\[
-5x = -16 \quad \Rightarrow \quad x = \frac{16}{5} = 3.2
\]
Thus, the zeros of the function are:
\[
x = 0 \quad \text{and} \quad x = 3.2
\]
In coordinate form, these zeros can be represented as points:
\((0, 0)\) and \((3.2, 0)\).
Therefore, the correct response is:
**(0,0)(3.2,0)**
**left parenthesis 0 comma 0 right parenthesis left parenthesis 3.2 comma 0 right parenthesis**
Answered by
GPT-4o mini
The given quadratic function is:
\[
y = 4(x - 25)^2 - 61
\]
This is in the vertex form of a parabola, which is represented as:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.
From your equation:
- \(h = 25\) (the value that \(x\) is shifted by)
- \(k = -61\) (the vertical shift)
Thus, the vertex of the parabola is:
\[
(25, -61)
\]
Therefore, the correct response is:
**(25,−61)**
**left parenthesis 25 comma negative 61 right parenthesis**
\[
y = 4(x - 25)^2 - 61
\]
This is in the vertex form of a parabola, which is represented as:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola.
From your equation:
- \(h = 25\) (the value that \(x\) is shifted by)
- \(k = -61\) (the vertical shift)
Thus, the vertex of the parabola is:
\[
(25, -61)
\]
Therefore, the correct response is:
**(25,−61)**
**left parenthesis 25 comma negative 61 right parenthesis**
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