Let's solve the problem step by step.
### Step (i): Expressing the Second Number
First, let's denote the two consecutive natural numbers as [tex]\( x \)[/tex] and [tex]\( x + 1 \)[/tex].
### Step (ii): Finding the Numbers
We know that the product of these two numbers is 30:
[tex]\[ x \times (x + 1) = 30 \][/tex]
This equation can be expanded and rewritten as:
[tex]\[ x^2 + x - 30 = 0 \][/tex]
This is a quadratic equation in standard form ([tex]\( ax^2 + bx + c = 0 \)[/tex]). We can solve it using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For our equation [tex]\( x^2 + x - 30 = 0 \)[/tex], the coefficients are [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -30 \)[/tex].
Plug these values into the quadratic formula:
[tex]\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-30)}}{2 \cdot 1} \][/tex]
Simplifying inside the square root:
[tex]\[ x = \frac{-1 \pm \sqrt{1 + 120}}{2} \][/tex]
[tex]\[ x = \frac{-1 \pm \sqrt{121}}{2} \][/tex]
[tex]\[ x = \frac{-1 \pm 11}{2} \][/tex]
This gives us two solutions:
[tex]\[ x = \frac{-1 + 11}{2} = \frac{10}{2} = 5 \][/tex]
[tex]\[ x = \frac{-1 - 11}{2} = \frac{-12}{2} = -6 \][/tex]
Since [tex]\( x \)[/tex] represents a natural number, we discard the negative solution. So, [tex]\( x = 5 \)[/tex].
Therefore, the two consecutive natural numbers are:
[tex]\[ x = 5 \][/tex]
[tex]\[ x + 1 = 6 \][/tex]
### Step (iii): Finding the Number to Add to the Smaller Number
We need to determine the number [tex]\( z \)[/tex] that should be added to the smaller number (which is 5) such that the sum of the squares of the two numbers becomes 85.
Let:
[tex]\[ 5 + z \][/tex]
[tex]\[ 6 \][/tex]
The sum of the squares of these two numbers should be equal to 85:
[tex]\[ (5 + z)^2 + 6^2 = 85 \][/tex]
Let's solve this equation step by step:
[tex]\[ (5 + z)^2 + 36 = 85 \][/tex]
[tex]\[ (5 + z)^2 = 85 - 36 \][/tex]
[tex]\[ (5 + z)^2 = 49 \][/tex]
Taking the square root of both sides:
[tex]\[ 5 + z = \pm \sqrt{49} \][/tex]
[tex]\[ 5 + z = \pm 7 \][/tex]
This gives us two solutions:
[tex]\[ 5 + z = 7 \][/tex]
[tex]\[ z = 2 \][/tex]
[tex]\[ 5 + z = -7 \][/tex]
[tex]\[ z = -12 \][/tex]
Since [tex]\( z \)[/tex] represents an additive number and typically, we would consider only non-negative values for such contexts, we take:
[tex]\[ z = 2 \][/tex]
### Conclusion:
1. The two consecutive natural numbers are 5 and 6.
2. The number [tex]\( z \)[/tex] that should be added to the smaller number (5) to make the sum of the squares of the two numbers 85 is [tex]\( z = 2 \)[/tex].
So, the final answer is:
- The two consecutive natural numbers are 5 and 6.
- The number to be added to 5 is 2.
