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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.
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