Sure! Let's go step by step to solve this problem.
### Given:
a) The sum of two numbers is 10 and the sum of their squares is 68.
#### (i) If one of the numbers is [tex]\( x \)[/tex], write the other number in terms of [tex]\( x \)[/tex].
If one of the numbers is [tex]\( x \)[/tex], then the other number can be written as [tex]\( 10 - x \)[/tex], since the sum of the two numbers is 10.
#### (ii) Find the numbers.
We need two equations here:
1. The sum of the numbers:
[tex]\[ x + (10 - x) = 10 \][/tex]
2. The sum of their squares:
[tex]\[ x^2 + (10 - x)^2 = 68 \][/tex]
Let's simplify the second equation step-by-step:
[tex]\[ x^2 + (10 - x)^2 = 68 \][/tex]
Expand [tex]\( (10 - x)^2 \)[/tex]:
[tex]\[ (10 - x)^2 = 100 - 20x + x^2 \][/tex]
So the equation becomes:
[tex]\[ x^2 + 100 - 20x + x^2 = 68 \][/tex]
Combine like terms:
[tex]\[ 2x^2 - 20x + 100 = 68 \][/tex]
Subtract 68 from both sides to set the equation to zero:
[tex]\[ 2x^2 - 20x + 32 = 0 \][/tex]
This is a quadratic equation [tex]\( 2x^2 - 20x + 32 = 0 \)[/tex].
To solve the quadratic equation, we use the quadratic formula [tex]\( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \)[/tex], where [tex]\( a = 2 \)[/tex], [tex]\( b = -20 \)[/tex], and [tex]\( c = 32 \)[/tex].
Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-20)^2 - 4 \cdot 2 \cdot 32 = 400 - 256 = 144 \][/tex]
Now find the roots:
[tex]\[ x = \frac{{20 \pm \sqrt{144}}}{4} \][/tex]
[tex]\[ x = \frac{{20 \pm 12}}{4} \][/tex]
So, we get two solutions:
[tex]\[ x_1 = \frac{{20 + 12}}{4} = \frac{32}{4} = 8 \][/tex]
[tex]\[ x_2 = \frac{{20 - 12}}{4} = \frac{8}{4} = 2 \][/tex]
The numbers are 8 and 2.
#### (iii) Find the ratio of the sum to the product of the numbers.
The sum of the numbers is:
[tex]\[ 8 + 2 = 10 \][/tex]
The product of the numbers is:
[tex]\[ 8 \times 2 = 16 \][/tex]
The ratio of the sum to the product is:
[tex]\[ \frac{{\text{sum of the numbers}}}{{\text{product of the numbers}}} = \frac{10}{16} = 0.625 \][/tex]
So, the ratio of the sum to the product of the numbers is [tex]\( 0.625 \)[/tex].
### Summary:
1. The numbers are 8 and 2.
2. The ratio of the sum to the product of the numbers is [tex]\( 0.625 \)[/tex].
