At Westonci.ca, we make it easy to get the answers you need from a community of informed and experienced contributors. Our platform offers a seamless experience for finding reliable answers from a network of experienced professionals. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
Certainly! Let's solve each of these questions step-by-step.
### 6. Factorize: [tex]\(2(x-y)^2-9(x-y)+10\)[/tex]
To factorize the given quadratic expression, we will employ substitution to simplify it.
1. Substitution:
Let [tex]\(u = (x - y)\)[/tex]. Then the expression [tex]\(2(x-y)^2 - 9(x-y) + 10\)[/tex] transforms to:
[tex]\[ 2u^2 - 9u + 10 \][/tex]
2. Quadratic Factorization:
To factorize [tex]\(2u^2 - 9u + 10\)[/tex], we find the roots using the quadratic formula:
[tex]\[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
- Here, [tex]\(a = 2\)[/tex], [tex]\(b = -9\)[/tex], and [tex]\(c = 10\)[/tex].
3. Calculate the Discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac = (-9)^2 - 4 \cdot 2 \cdot 10 \][/tex]
[tex]\[ = 81 - 80 = 1 \][/tex]
4. Find the Roots:
[tex]\[ u_1 = \frac{-(-9) + \sqrt{1}}{2 \cdot 2} = \frac{9 + 1}{4} = \frac{10}{4} = 2.5 \][/tex]
[tex]\[ u_2 = \frac{-(-9) - \sqrt{1}}{2 \cdot 2} = \frac{9 - 1}{4} = \frac{8}{4} = 2 \][/tex]
5. Factor Form:
So, the quadratic [tex]\(2u^2 - 9u + 10\)[/tex] can be factored into:
[tex]\[ 2(u - 2.5)(u - 2) \][/tex]
6. Substitute Back [tex]\(u = (x - y)\)[/tex]:
[tex]\[ 2((x - y) - 2.5)((x - y) - 2) \][/tex]
Therefore, the factorized form of the expression [tex]\(2(x-y)^2 - 9(x-y) + 10\)[/tex] is:
[tex]\[ 2((x-y) - 2.5)((x-y) - 2) \][/tex]
### 7. Set Operations
#### i) List the elements of the sets [tex]\(X\)[/tex], [tex]\(Y\)[/tex], and [tex]\(Z\)[/tex].
- Set [tex]\( X \)[/tex]: Multiples of 2 up to 20.
[tex]\[ X = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\} \][/tex]
- Set [tex]\( Y \)[/tex]: Even numbers up to 10.
[tex]\[ Y = \{2, 4, 6, 8, 10\} \][/tex]
- Set [tex]\( Z \)[/tex]: Factors of 16 greater than 1.
[tex]\[ Z = \{2, 4, 8, 16\} \][/tex]
#### ii) Are the subsets proper and improper? Give Reason.
- Is [tex]\(Y\)[/tex] a subset of [tex]\(X\)[/tex]?
[tex]\[ Y \subseteq X \][/tex]
All elements of [tex]\(Y\)[/tex] are in [tex]\(X\)[/tex]. Therefore, [tex]\(Y\)[/tex] is a subset of [tex]\(X\)[/tex]. Since [tex]\(Y\)[/tex] does not contain all elements of [tex]\(X\)[/tex] and [tex]\(X \neq Y\)[/tex], it is a proper subset.
- Is [tex]\(Z\)[/tex] a subset of [tex]\(X\)[/tex]?
[tex]\[ Z \subseteq X \][/tex]
All elements of [tex]\(Z\)[/tex] are in [tex]\(X\)[/tex]. Therefore, [tex]\(Z\)[/tex] is a subset of [tex]\(X\)[/tex]. Since [tex]\(Z\)[/tex] does not contain all elements of [tex]\(X\)[/tex] and [tex]\(X \neq Z\)[/tex], it is a proper subset.
#### iii) Combine the elements of [tex]\(Y\)[/tex] and [tex]\(Z\)[/tex] and show it in the Venn Diagram along with the universal set.
- Union of [tex]\(Y\)[/tex] and [tex]\(Z\)[/tex]:
[tex]\[ Y \cup Z = \{2, 4, 6, 8, 10, 16\} \][/tex]
- Universal Set:
Let's consider the universal set up to 20:
[tex]\[ \text{Universal Set} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\} \][/tex]
### Venn Diagram
1. Universal Set: [tex]\(\{1, 2, 3, \dots, 20\}\)[/tex]
2. Set [tex]\(X\)[/tex]: [tex]\(\{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}\)[/tex]
3. Set [tex]\(Y\)[/tex]: [tex]\(\{2, 4, 6, 8, 10\}\)[/tex]
4. Set [tex]\(Z\)[/tex]: [tex]\(\{2, 4, 8, 16\}\)[/tex]
5. Union [tex]\(Y \cup Z\)[/tex]: [tex]\(\{2, 4, 6, 8, 10, 16\}\)[/tex]
In a Venn diagram, we can represent the relationship between [tex]\(X\)[/tex], [tex]\(Y\)[/tex], [tex]\(Z\)[/tex], and their union within the universal set:
- [tex]\(X\)[/tex] will encompass all the multiples of 2 up to 20.
- [tex]\(Y\)[/tex] will be within [tex]\(X\)[/tex] containing the smaller set.
- [tex]\(Z\)[/tex] will overlap with [tex]\(Y\)[/tex] since they share common elements.
- Union [tex]\(Y \cup Z\)[/tex] combines both.
Unfortunately, a text-based Venn diagram isn't possible here, but the relationships can be understood from the element descriptions.
### 6. Factorize: [tex]\(2(x-y)^2-9(x-y)+10\)[/tex]
To factorize the given quadratic expression, we will employ substitution to simplify it.
1. Substitution:
Let [tex]\(u = (x - y)\)[/tex]. Then the expression [tex]\(2(x-y)^2 - 9(x-y) + 10\)[/tex] transforms to:
[tex]\[ 2u^2 - 9u + 10 \][/tex]
2. Quadratic Factorization:
To factorize [tex]\(2u^2 - 9u + 10\)[/tex], we find the roots using the quadratic formula:
[tex]\[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
- Here, [tex]\(a = 2\)[/tex], [tex]\(b = -9\)[/tex], and [tex]\(c = 10\)[/tex].
3. Calculate the Discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac = (-9)^2 - 4 \cdot 2 \cdot 10 \][/tex]
[tex]\[ = 81 - 80 = 1 \][/tex]
4. Find the Roots:
[tex]\[ u_1 = \frac{-(-9) + \sqrt{1}}{2 \cdot 2} = \frac{9 + 1}{4} = \frac{10}{4} = 2.5 \][/tex]
[tex]\[ u_2 = \frac{-(-9) - \sqrt{1}}{2 \cdot 2} = \frac{9 - 1}{4} = \frac{8}{4} = 2 \][/tex]
5. Factor Form:
So, the quadratic [tex]\(2u^2 - 9u + 10\)[/tex] can be factored into:
[tex]\[ 2(u - 2.5)(u - 2) \][/tex]
6. Substitute Back [tex]\(u = (x - y)\)[/tex]:
[tex]\[ 2((x - y) - 2.5)((x - y) - 2) \][/tex]
Therefore, the factorized form of the expression [tex]\(2(x-y)^2 - 9(x-y) + 10\)[/tex] is:
[tex]\[ 2((x-y) - 2.5)((x-y) - 2) \][/tex]
### 7. Set Operations
#### i) List the elements of the sets [tex]\(X\)[/tex], [tex]\(Y\)[/tex], and [tex]\(Z\)[/tex].
- Set [tex]\( X \)[/tex]: Multiples of 2 up to 20.
[tex]\[ X = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\} \][/tex]
- Set [tex]\( Y \)[/tex]: Even numbers up to 10.
[tex]\[ Y = \{2, 4, 6, 8, 10\} \][/tex]
- Set [tex]\( Z \)[/tex]: Factors of 16 greater than 1.
[tex]\[ Z = \{2, 4, 8, 16\} \][/tex]
#### ii) Are the subsets proper and improper? Give Reason.
- Is [tex]\(Y\)[/tex] a subset of [tex]\(X\)[/tex]?
[tex]\[ Y \subseteq X \][/tex]
All elements of [tex]\(Y\)[/tex] are in [tex]\(X\)[/tex]. Therefore, [tex]\(Y\)[/tex] is a subset of [tex]\(X\)[/tex]. Since [tex]\(Y\)[/tex] does not contain all elements of [tex]\(X\)[/tex] and [tex]\(X \neq Y\)[/tex], it is a proper subset.
- Is [tex]\(Z\)[/tex] a subset of [tex]\(X\)[/tex]?
[tex]\[ Z \subseteq X \][/tex]
All elements of [tex]\(Z\)[/tex] are in [tex]\(X\)[/tex]. Therefore, [tex]\(Z\)[/tex] is a subset of [tex]\(X\)[/tex]. Since [tex]\(Z\)[/tex] does not contain all elements of [tex]\(X\)[/tex] and [tex]\(X \neq Z\)[/tex], it is a proper subset.
#### iii) Combine the elements of [tex]\(Y\)[/tex] and [tex]\(Z\)[/tex] and show it in the Venn Diagram along with the universal set.
- Union of [tex]\(Y\)[/tex] and [tex]\(Z\)[/tex]:
[tex]\[ Y \cup Z = \{2, 4, 6, 8, 10, 16\} \][/tex]
- Universal Set:
Let's consider the universal set up to 20:
[tex]\[ \text{Universal Set} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\} \][/tex]
### Venn Diagram
1. Universal Set: [tex]\(\{1, 2, 3, \dots, 20\}\)[/tex]
2. Set [tex]\(X\)[/tex]: [tex]\(\{2, 4, 6, 8, 10, 12, 14, 16, 18, 20\}\)[/tex]
3. Set [tex]\(Y\)[/tex]: [tex]\(\{2, 4, 6, 8, 10\}\)[/tex]
4. Set [tex]\(Z\)[/tex]: [tex]\(\{2, 4, 8, 16\}\)[/tex]
5. Union [tex]\(Y \cup Z\)[/tex]: [tex]\(\{2, 4, 6, 8, 10, 16\}\)[/tex]
In a Venn diagram, we can represent the relationship between [tex]\(X\)[/tex], [tex]\(Y\)[/tex], [tex]\(Z\)[/tex], and their union within the universal set:
- [tex]\(X\)[/tex] will encompass all the multiples of 2 up to 20.
- [tex]\(Y\)[/tex] will be within [tex]\(X\)[/tex] containing the smaller set.
- [tex]\(Z\)[/tex] will overlap with [tex]\(Y\)[/tex] since they share common elements.
- Union [tex]\(Y \cup Z\)[/tex] combines both.
Unfortunately, a text-based Venn diagram isn't possible here, but the relationships can be understood from the element descriptions.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.
