Answer :
Sure! Let's break this problem down step-by-step.
### (i) If one of the numbers is x, write another number in terms of x.
If we have two consecutive even numbers and we denote the first one by [tex]\( x \)[/tex], then the next consecutive even number will be [tex]\( x + 2 \)[/tex].
So, if one of the numbers is [tex]\( x \)[/tex], the other number in terms of [tex]\( x \)[/tex] will be [tex]\( x + 2 \)[/tex].
### (ii) Find the numbers.
We are given that the product of the two consecutive even numbers is 80. Therefore, we can write the equation:
[tex]\[ x \times (x + 2) = 80 \][/tex]
Expanding this equation:
[tex]\[ x^2 + 2x = 80 \][/tex]
To solve for [tex]\( x \)[/tex], we first rearrange the equation:
[tex]\[ x^2 + 2x - 80 = 0 \][/tex]
This is a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -80 \)[/tex].
Solving this quadratic equation, we find that:
[tex]\[ x = 8 \][/tex]
Thus, the two consecutive even numbers are:
[tex]\[ x = 8 \][/tex]
[tex]\[ x + 2 = 10 \][/tex]
So, the two consecutive even numbers are 8 and 10.
### (iii) What is the number that must be added to the smaller one and subtracted from the greater number so that the product of the two numbers is 777?
We need to find a number [tex]\( a \)[/tex] such that when it is added to the smaller number (8) and subtracted from the greater number (10), their product is 777. Let's denote the smaller number by [tex]\( 8 \)[/tex] and the greater number by [tex]\( 10 \)[/tex]. We need:
[tex]\[ (8 + a) \times (10 - a) = 777 \][/tex]
Expanding this equation:
[tex]\[ 80 - 8a + 10a - a^2 = 777 \][/tex]
[tex]\[ -a^2 + 2a + 80 = 777 \][/tex]
Rearranging to be in standard quadratic form:
[tex]\[ -a^2 + 2a + 80 - 777 = 0 \][/tex]
[tex]\[ -a^2 + 2a - 697 = 0 \][/tex]
Solving this quadratic equation for [tex]\( a \)[/tex], we find the solutions are:
[tex]\[ a = 1 - 2\sqrt{174}i \][/tex]
[tex]\[ a = 1 + 2\sqrt{174}i \][/tex]
Since the solutions for [tex]\( a \)[/tex] involve imaginary numbers, it implies that there are no real values for [tex]\( a \)[/tex] that satisfy the equation given. Thus, the number that must be added to the smaller one and subtracted from the greater one to get their product as 777 is not a real number.
### (i) If one of the numbers is x, write another number in terms of x.
If we have two consecutive even numbers and we denote the first one by [tex]\( x \)[/tex], then the next consecutive even number will be [tex]\( x + 2 \)[/tex].
So, if one of the numbers is [tex]\( x \)[/tex], the other number in terms of [tex]\( x \)[/tex] will be [tex]\( x + 2 \)[/tex].
### (ii) Find the numbers.
We are given that the product of the two consecutive even numbers is 80. Therefore, we can write the equation:
[tex]\[ x \times (x + 2) = 80 \][/tex]
Expanding this equation:
[tex]\[ x^2 + 2x = 80 \][/tex]
To solve for [tex]\( x \)[/tex], we first rearrange the equation:
[tex]\[ x^2 + 2x - 80 = 0 \][/tex]
This is a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -80 \)[/tex].
Solving this quadratic equation, we find that:
[tex]\[ x = 8 \][/tex]
Thus, the two consecutive even numbers are:
[tex]\[ x = 8 \][/tex]
[tex]\[ x + 2 = 10 \][/tex]
So, the two consecutive even numbers are 8 and 10.
### (iii) What is the number that must be added to the smaller one and subtracted from the greater number so that the product of the two numbers is 777?
We need to find a number [tex]\( a \)[/tex] such that when it is added to the smaller number (8) and subtracted from the greater number (10), their product is 777. Let's denote the smaller number by [tex]\( 8 \)[/tex] and the greater number by [tex]\( 10 \)[/tex]. We need:
[tex]\[ (8 + a) \times (10 - a) = 777 \][/tex]
Expanding this equation:
[tex]\[ 80 - 8a + 10a - a^2 = 777 \][/tex]
[tex]\[ -a^2 + 2a + 80 = 777 \][/tex]
Rearranging to be in standard quadratic form:
[tex]\[ -a^2 + 2a + 80 - 777 = 0 \][/tex]
[tex]\[ -a^2 + 2a - 697 = 0 \][/tex]
Solving this quadratic equation for [tex]\( a \)[/tex], we find the solutions are:
[tex]\[ a = 1 - 2\sqrt{174}i \][/tex]
[tex]\[ a = 1 + 2\sqrt{174}i \][/tex]
Since the solutions for [tex]\( a \)[/tex] involve imaginary numbers, it implies that there are no real values for [tex]\( a \)[/tex] that satisfy the equation given. Thus, the number that must be added to the smaller one and subtracted from the greater one to get their product as 777 is not a real number.
