Question
Order the steps from first to last that would be used to factor:
3x^2−13x−10
3x^2−13x−10
Answers
Answer
Fill in the blank to show what the quadratic expression would look like when factored.(5 points)
x^2−3x−10= (x− Response area)(x+ Response area)
3x^2−6x+9 = Response area (x^2−2x+3)
9x^2−25= (3x− Response area)(Response area+5)
x^2−3x−10= (x− Response area)(x+ Response area)
3x^2−6x+9 = Response area (x^2−2x+3)
9x^2−25= (3x− Response area)(Response area+5)
Answer
there is no -5 only 5
Answer
What are the x-intercepts of the quadratic equation below?
y=(2x−3)(x−6)
(1 point)
Responses
(-1.5, 0) and (-6, 0)
(1.5, 0) and (6, 0)
(3, 0) and (6, 0)
(-3, 0) and (-6, 0)
y=(2x−3)(x−6)
(1 point)
Responses
(-1.5, 0) and (-6, 0)
(1.5, 0) and (6, 0)
(3, 0) and (6, 0)
(-3, 0) and (-6, 0)
Answer
What is the minimum of the graph described by y=(x+4)^2 −5
(1 point)
Responses
(-4, 5)
(-4, -5)
(4, -5)
(4, 5)
(1 point)
Responses
(-4, 5)
(-4, -5)
(4, -5)
(4, 5)
Answer
A rock is thrown from one side of a river to another. The function h(t) = −16t2 + 80t + 30
gives the height in inches of the rock t seconds after it has been thrown.(3 points)
a) Find h(2.5). Response area
b) What does h(2.5) represent in the context of this problem? Response area
c) Does it make sense to find h(−10) in the context of this problem? Response area
gives the height in inches of the rock t seconds after it has been thrown.(3 points)
a) Find h(2.5). Response area
b) What does h(2.5) represent in the context of this problem? Response area
c) Does it make sense to find h(−10) in the context of this problem? Response area
Answered by
GPT-4o mini
To factor the quadratic expression \(3x^2 - 13x - 10\), you can follow these steps:
1. **Identify**: Start with the quadratic in the form \(ax^2 + bx + c\), where \(a = 3\), \(b = -13\), and \(c = -10\).
2. **Multiply \(a\) and \(c\)**: Calculate \(a \cdot c\):
\[
3 \cdot (-10) = -30.
\]
3. **Find two numbers that multiply to \(ac\) and add to \(b\)**: Look for two numbers that multiply to \(-30\) and add to \(-13\). The numbers \(-15\) and \(2\) work because:
\[
-15 \cdot 2 = -30 \quad \text{and} \quad -15 + 2 = -13.
\]
4. **Rewrite the middle term**: Rewrite the expression using the two numbers found:
\[
3x^2 - 15x + 2x - 10.
\]
5. **Group the terms**: Group the terms into two pairs:
\[
(3x^2 - 15x) + (2x - 10).
\]
6. **Factor out the common factors from each group**:
\[
3x(x - 5) + 2(x - 5).
\]
7. **Factor out the common binomial**:
\[
(3x + 2)(x - 5).
\]
So, the factored form of the expression \(3x^2 - 13x - 10\) is:
\[
(3x + 2)(x - 5).
\]
Thus, the steps from first to last are:
1. Identify \(a\), \(b\), and \(c\).
2. Multiply \(a\) and \(c\).
3. Find two numbers that multiply to \(ac\) and add to \(b\).
4. Rewrite the middle term using these two numbers.
5. Group the terms.
6. Factor out the common factors from each group.
7. Factor out the common binomial.
1. **Identify**: Start with the quadratic in the form \(ax^2 + bx + c\), where \(a = 3\), \(b = -13\), and \(c = -10\).
2. **Multiply \(a\) and \(c\)**: Calculate \(a \cdot c\):
\[
3 \cdot (-10) = -30.
\]
3. **Find two numbers that multiply to \(ac\) and add to \(b\)**: Look for two numbers that multiply to \(-30\) and add to \(-13\). The numbers \(-15\) and \(2\) work because:
\[
-15 \cdot 2 = -30 \quad \text{and} \quad -15 + 2 = -13.
\]
4. **Rewrite the middle term**: Rewrite the expression using the two numbers found:
\[
3x^2 - 15x + 2x - 10.
\]
5. **Group the terms**: Group the terms into two pairs:
\[
(3x^2 - 15x) + (2x - 10).
\]
6. **Factor out the common factors from each group**:
\[
3x(x - 5) + 2(x - 5).
\]
7. **Factor out the common binomial**:
\[
(3x + 2)(x - 5).
\]
So, the factored form of the expression \(3x^2 - 13x - 10\) is:
\[
(3x + 2)(x - 5).
\]
Thus, the steps from first to last are:
1. Identify \(a\), \(b\), and \(c\).
2. Multiply \(a\) and \(c\).
3. Find two numbers that multiply to \(ac\) and add to \(b\).
4. Rewrite the middle term using these two numbers.
5. Group the terms.
6. Factor out the common factors from each group.
7. Factor out the common binomial.
Answered by
GPT-4o mini
Let's fill in the blanks for the provided quadratic expressions:
1. **For \(x^2 - 3x - 10\)**:
We need to factor this quadratic. We need two numbers that multiply to \(-10\) and add to \(-3\). The numbers are \(-5\) and \(2\).
So, the factors are:
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]
**Response areas**: \(-5\) and \(2\).
2. **For \(3x^2 - 6x + 9\)**:
First, we can factor out the greatest common factor (GCF), which is \(3\):
\[
3x^2 - 6x + 9 = 3(x^2 - 2x + 3)
\]
Thus, the response area is:
\[
3
\]
**Response area**: \(3\).
3. **For \(9x^2 - 25\)**:
This is a difference of squares, which factors as:
\[
9x^2 - 25 = (3x - 5)(3x + 5).
\]
Thus, we have \(3x - 5\) and \(3x + 5\).
So, the response areas are:
\[
5 \quad \text{and} \quad 5.
\]
**Response areas**: \(5\) and \(5\).
### Final Response Areas:
1. \((x - 5)(x + 2)\) with response areas: **-5** and **2**.
2. \(3(x^2 - 2x + 3)\) with response area: **3**.
3. \((3x - 5)(3x + 5)\) with response areas: **5** and **5**.
1. **For \(x^2 - 3x - 10\)**:
We need to factor this quadratic. We need two numbers that multiply to \(-10\) and add to \(-3\). The numbers are \(-5\) and \(2\).
So, the factors are:
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]
**Response areas**: \(-5\) and \(2\).
2. **For \(3x^2 - 6x + 9\)**:
First, we can factor out the greatest common factor (GCF), which is \(3\):
\[
3x^2 - 6x + 9 = 3(x^2 - 2x + 3)
\]
Thus, the response area is:
\[
3
\]
**Response area**: \(3\).
3. **For \(9x^2 - 25\)**:
This is a difference of squares, which factors as:
\[
9x^2 - 25 = (3x - 5)(3x + 5).
\]
Thus, we have \(3x - 5\) and \(3x + 5\).
So, the response areas are:
\[
5 \quad \text{and} \quad 5.
\]
**Response areas**: \(5\) and \(5\).
### Final Response Areas:
1. \((x - 5)(x + 2)\) with response areas: **-5** and **2**.
2. \(3(x^2 - 2x + 3)\) with response area: **3**.
3. \((3x - 5)(3x + 5)\) with response areas: **5** and **5**.
Answered by
GPT-4o mini
Thank you for the clarification! Let's go through the expressions again to ensure accuracy and align with your requirements.
1. **For \(x^2 - 3x - 10\)**:
- We need to factor this quadratic expression. The numbers that multiply to \(-10\) and add to \(-3\) are \(2\) and \(-5\).
- So the factors will be:
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]
- Therefore, the **Response areas** are: **5** and **2** (with \(5\) coming from \(x - 5\)).
2. **For \(3x^2 - 6x + 9\)**:
- We can factor out the greatest common factor (GCF), which is \(3\):
\[
3x^2 - 6x + 9 = 3(x^2 - 2x + 3)
\]
- Thus, the **Response area** is: **3**.
3. **For \(9x^2 - 25\)**:
- This is a difference of squares:
\[
9x^2 - 25 = (3x - 5)(3x + 5).
\]
- Therefore, the **Response areas** are \(5\) and \(5\).
### Final Filled Responses:
1. \((x - 5)(x + 2)\) with response areas: **5** and **2**.
2. \(3(x^2 - 2x + 3)\) with response area: **3**.
3. \((3x - 5)(3x + 5)\) with response areas: **5** and **5**.
Thank you for your patience, and I hope this accurately reflects your requirements!
1. **For \(x^2 - 3x - 10\)**:
- We need to factor this quadratic expression. The numbers that multiply to \(-10\) and add to \(-3\) are \(2\) and \(-5\).
- So the factors will be:
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]
- Therefore, the **Response areas** are: **5** and **2** (with \(5\) coming from \(x - 5\)).
2. **For \(3x^2 - 6x + 9\)**:
- We can factor out the greatest common factor (GCF), which is \(3\):
\[
3x^2 - 6x + 9 = 3(x^2 - 2x + 3)
\]
- Thus, the **Response area** is: **3**.
3. **For \(9x^2 - 25\)**:
- This is a difference of squares:
\[
9x^2 - 25 = (3x - 5)(3x + 5).
\]
- Therefore, the **Response areas** are \(5\) and \(5\).
### Final Filled Responses:
1. \((x - 5)(x + 2)\) with response areas: **5** and **2**.
2. \(3(x^2 - 2x + 3)\) with response area: **3**.
3. \((3x - 5)(3x + 5)\) with response areas: **5** and **5**.
Thank you for your patience, and I hope this accurately reflects your requirements!
Answered by
GPT-4o mini
To find the x-intercepts of the quadratic equation given in factored form, \(y = (2x - 3)(x - 6)\), we set \(y\) to zero and solve for \(x\):
\[
0 = (2x - 3)(x - 6)
\]
This equation is satisfied when either factor is equal to zero.
1. Set the first factor equal to zero:
\[
2x - 3 = 0
\]
Solve for \(x\):
\[
2x = 3 \implies x = \frac{3}{2} = 1.5.
\]
2. Set the second factor equal to zero:
\[
x - 6 = 0
\]
Solve for \(x\):
\[
x = 6.
\]
So, the x-intercepts are at the points \((1.5, 0)\) and \((6, 0)\).
### Answer:
**(1.5, 0) and (6, 0)**.
\[
0 = (2x - 3)(x - 6)
\]
This equation is satisfied when either factor is equal to zero.
1. Set the first factor equal to zero:
\[
2x - 3 = 0
\]
Solve for \(x\):
\[
2x = 3 \implies x = \frac{3}{2} = 1.5.
\]
2. Set the second factor equal to zero:
\[
x - 6 = 0
\]
Solve for \(x\):
\[
x = 6.
\]
So, the x-intercepts are at the points \((1.5, 0)\) and \((6, 0)\).
### Answer:
**(1.5, 0) and (6, 0)**.
Answered by
GPT-4o mini
To find the minimum of the quadratic function given by
\[
y = (x + 4)^2 - 5,
\]
we can analyze its vertex form.
The expression \((x + 4)^2\) represents a parabola that opens upwards (since the coefficient of the squared term is positive). The vertex of this parabola is also its minimum point.
The vertex form of a quadratic function is given by
\[
y = a(x - h)^2 + k,
\]
where \((h, k)\) is the vertex. In this case:
- The term \((x + 4)\) can be rewritten as \((x - (-4))\), thus \(h = -4\).
- The value of \(k\) is \(-5\).
Therefore, the vertex (minimum point) of the graph is at
\[
(-4, -5).
\]
### Answer:
**(-4, -5)**.
\[
y = (x + 4)^2 - 5,
\]
we can analyze its vertex form.
The expression \((x + 4)^2\) represents a parabola that opens upwards (since the coefficient of the squared term is positive). The vertex of this parabola is also its minimum point.
The vertex form of a quadratic function is given by
\[
y = a(x - h)^2 + k,
\]
where \((h, k)\) is the vertex. In this case:
- The term \((x + 4)\) can be rewritten as \((x - (-4))\), thus \(h = -4\).
- The value of \(k\) is \(-5\).
Therefore, the vertex (minimum point) of the graph is at
\[
(-4, -5).
\]
### Answer:
**(-4, -5)**.
Answered by
GPT-4o mini
Let's analyze the function \( h(t) = -16t^2 + 80t + 30 \) and work through the questions step-by-step.
### a) Find \( h(2.5) \).
To find \( h(2.5) \), substitute \( t = 2.5 \) into the function:
\[
h(2.5) = -16(2.5)^2 + 80(2.5) + 30.
\]
Calculating each term:
1. Calculate \( (2.5)^2 \):
\[
(2.5)^2 = 6.25.
\]
2. Calculate \( -16(2.5)^2 \):
\[
-16 \times 6.25 = -100.
\]
3. Calculate \( 80(2.5) \):
\[
80 \times 2.5 = 200.
\]
4. Now add all the parts together:
\[
h(2.5) = -100 + 200 + 30 = 130.
\]
So, \( h(2.5) = 130 \).
**Response area**: **130**.
### b) What does \( h(2.5) \) represent in the context of this problem?
In this problem, \( h(t) \) represents the height of the rock in inches, \( t \) seconds after it has been thrown. Therefore, \( h(2.5) = 130 \) means that at 2.5 seconds after it was thrown, the height of the rock is **130 inches** above the ground.
**Response area**: **At 2.5 seconds, the height of the rock is 130 inches above the ground.**
### c) Does it make sense to find \( h(-10) \) in the context of this problem?
In the context of this problem, \( t \) represents time measured in seconds after the rock is thrown. Since time cannot be negative, \( h(-10) \) does not make sense in this context. It would imply looking for the height of the rock 10 seconds before it was thrown, which is not relevant to the situation being modeled.
**Response area**: **No, it does not make sense to find \( h(-10) \) since time cannot be negative.**
### a) Find \( h(2.5) \).
To find \( h(2.5) \), substitute \( t = 2.5 \) into the function:
\[
h(2.5) = -16(2.5)^2 + 80(2.5) + 30.
\]
Calculating each term:
1. Calculate \( (2.5)^2 \):
\[
(2.5)^2 = 6.25.
\]
2. Calculate \( -16(2.5)^2 \):
\[
-16 \times 6.25 = -100.
\]
3. Calculate \( 80(2.5) \):
\[
80 \times 2.5 = 200.
\]
4. Now add all the parts together:
\[
h(2.5) = -100 + 200 + 30 = 130.
\]
So, \( h(2.5) = 130 \).
**Response area**: **130**.
### b) What does \( h(2.5) \) represent in the context of this problem?
In this problem, \( h(t) \) represents the height of the rock in inches, \( t \) seconds after it has been thrown. Therefore, \( h(2.5) = 130 \) means that at 2.5 seconds after it was thrown, the height of the rock is **130 inches** above the ground.
**Response area**: **At 2.5 seconds, the height of the rock is 130 inches above the ground.**
### c) Does it make sense to find \( h(-10) \) in the context of this problem?
In the context of this problem, \( t \) represents time measured in seconds after the rock is thrown. Since time cannot be negative, \( h(-10) \) does not make sense in this context. It would imply looking for the height of the rock 10 seconds before it was thrown, which is not relevant to the situation being modeled.
**Response area**: **No, it does not make sense to find \( h(-10) \) since time cannot be negative.**