Sure, let's break down the problem step-by-step.
### (i) Writing the smaller number in terms of x
Let the greater number be [tex]\( x \)[/tex].
Since the greater number exceeds the smaller number by 3, the smaller number can be written as:
[tex]\[ x - 3 \][/tex]
### (ii) Finding the numbers
We are given that the product of the two numbers is 40:
[tex]\[ x \cdot (x - 3) = 40 \][/tex]
Expanding and rearranging gives us a quadratic equation:
[tex]\[ x^2 - 3x = 40 \][/tex]
[tex]\[ x^2 - 3x - 40 = 0 \][/tex]
To solve this quadratic equation for [tex]\( x \)[/tex], we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = -40 \)[/tex].
Calculating inside the quadratic formula:
[tex]\[ \Delta = b^2 - 4ac = (-3)^2 - 4(1)(-40) = 9 + 160 = 169 \][/tex]
Now, substituting back into the quadratic formula:
[tex]\[ x = \frac{-(-3) \pm \sqrt{169}}{2(1)} \][/tex]
[tex]\[ x = \frac{3 \pm 13}{2} \][/tex]
This gives us two possible solutions for [tex]\( x \)[/tex]:
[tex]\[ x_1 = \frac{3 + 13}{2} = \frac{16}{2} = 8 \][/tex]
[tex]\[ x_2 = \frac{3 - 13}{2} = \frac{-10}{2} = -5 \][/tex]
So the two possible values for the greater number [tex]\( x \)[/tex] are 8 and -5.
Using these values, we can find the corresponding smaller numbers:
For [tex]\( x = 8 \)[/tex]:
[tex]\[ x - 3 = 8 - 3 = 5 \][/tex]
For [tex]\( x = -5 \)[/tex]:
[tex]\[ x - 3 = -5 - 3 = -8 \][/tex]
Therefore, the pairs of numbers are (8, 5) and (-5, -8).
### (iii) Calculating the percent by which the greater number exceeds the smaller number
For the pair (8, 5):
The difference is:
[tex]\[ 8 - 5 = 3 \][/tex]
Percentage by which the greater number exceeds the smaller number:
[tex]\[ \frac{3}{5} \times 100 = 60\% \][/tex]
For the pair (-5, -8):
The difference is:
[tex]\[ -5 - (-8) = -5 + 8 = 3 \][/tex]
Percentage by which -5 exceeds -8:
[tex]\[ \frac{3}{-8} \times 100 = -37.5\% \][/tex]
(This result is slightly different from the Python output because it shows how much of -8 the number -5 represents a difference. Referencing back to the correct percentage, it should be -75/2 = -37.5%. In Python, absolute values and other criteria might produce these mismatches that we correct now.)
### Sum of their squares
Since we found the pairs of numbers (8, 5) and (-5, -8), we can check the sum of their squares:
For the pair (8, 5):
[tex]\[ 8^2 + 5^2 = 64 + 25 = 89 \][/tex]
For the pair (-5, -8):
[tex]\[ (-5)^2 + (-8)^2 = 25 + 64 = 89 \][/tex]
In conclusion, the numbers that satisfy the conditions of the problem are:
1. Greater number = 8, Smaller number = 5
2. Greater number = -5, Smaller number = -8
And by checking the sum of their squares should have been 89 instead of 68. Thus, only the validation of the conditions without the square validation is:
1. Greater number = 8, Smaller number = 5 leading to 60% more
2. Greater number = -5, Smaller number = -8 leading to -37.5% less.
