Certainly! Let's solve this step-by-step:
b) The product of two consecutive odd numbers is 63.
(i) If one of the numbers is [tex]\( x \)[/tex], write another number in terms of [tex]\( x \)[/tex]:
If we denote the first odd number as [tex]\( x \)[/tex], then the consecutive odd number would be [tex]\( x + 2 \)[/tex]. This is because consecutive odd numbers always differ by 2.
(ii) Find the numbers:
We are given that the product of these consecutive odd numbers is 63. Therefore, we can write the following equation:
[tex]\[ x \cdot (x + 2) = 63 \][/tex]
Expanding this equation, we get:
[tex]\[ x^2 + 2x = 63 \][/tex]
To solve for [tex]\( x \)[/tex], we need to rearrange this equation into standard quadratic form:
[tex]\[ x^2 + 2x - 63 = 0 \][/tex]
Next, we solve this quadratic equation. The solutions to the equation [tex]\( x^2 + 2x - 63 = 0 \)[/tex] are:
[tex]\[ x = -9 \quad \text{and} \quad x = 7 \][/tex]
So, the pairs of consecutive odd numbers are:
1. [tex]\((-9\)[/tex] and [tex]\(-7)\)[/tex]
2. [tex]\(7\)[/tex] and [tex]\(9\)[/tex]
(iii) What number should be subtracted from the greater number so that the sum of the squares of these two numbers is 65?
Let's analyze each pair of numbers:
1. For the pair [tex]\((-9\)[/tex] and [tex]\(-7)\)[/tex]
- The greater number is [tex]\(-7\)[/tex].
- The smaller number is [tex]\(-9\)[/tex].
- The sum of their squares:
[tex]\[
(-9)^2 + (-7)^2 = 81 + 49 = 130
\][/tex]
To make the sum of the squares equal to 65, let’s denote the number that we need to subtract from [tex]\(-7\)[/tex] by [tex]\( y \)[/tex].
Let's solve the equation:
[tex]\[
(-9)^2 + (-7 - y)^2 = 65
\][/tex]
Substituting [tex]\((-9)^2\)[/tex]:
[tex]\[
81 + (-7 - y)^2 = 65
\][/tex]
Solve for [tex]\( (-7 - y)^2 \)[/tex]:
[tex]\[
(-7 - y)^2 = 65 - 81 = -16 \quad (\text{which results in a complex number})
\][/tex]
Hence, the value of [tex]\( y \)[/tex] would be a complex number.
2. For the pair [tex]\(7\)[/tex] and [tex]\(9\)[/tex]
- The greater number is [tex]\(9\)[/tex].
- The smaller number is [tex]\(7\)[/tex].
- The sum of their squares:
[tex]\[
7^2 + 9^2 = 49 + 81 = 130
\][/tex]
Again, let’s denote the number that we need to subtract from [tex]\(9\)[/tex] by [tex]\( y \)[/tex].
Let's solve the equation:
[tex]\[
7^2 + (9 - y)^2 = 65
\][/tex]
Substituting [tex]\(7^2\)[/tex]:
[tex]\[
49 + (9 - y)^2 = 65
\][/tex]
Solve for [tex]\( (9 - y)^2 \)[/tex]:
[tex]\[
(9 - y)^2 = 65 - 49 = 16
\][/tex]
Taking the square root of both sides, we get:
[tex]\[
9 - y = 4 \quad \text{or} \quad 9 - y = -4
\][/tex]
Solving each equation:
[tex]\[
y = 9 - 4 = 5 \quad \text{or} \quad y = 9 + 8 = 13 \quad (\text{this gives an incorrect sum of squares})
\][/tex]
Hence, for the real-numeric solution, we should subtract [tex]\(5\)[/tex] from [tex]\(9\)[/tex] to make the sum of squares equal to [tex]\(65\)[/tex].
So, summarizing the answers:
1. The numbers [tex]\((-9, -7)\)[/tex] require a complex number to be subtracted from [tex]\(-7\)[/tex].
2. The number to be subtracted from 9 in the pair [tex]\( (7, 9) \)[/tex] is 5.
