To solve the problem, we need to determine two consecutive multiples of 8 whose product is 768. Let's break this down step by step.
### Step 1: Define the Variables
Let the first multiple of 8 be [tex]\(8n\)[/tex]. Therefore, the next consecutive multiple of 8 will be [tex]\(8(n + 1)\)[/tex].
### Step 2: Set Up the Product Equation
The product of these two multiples is given to be 768:
[tex]\[ (8n) \times 8(n + 1) = 768 \][/tex]
### Step 3: Simplify the Equation
Simplify the left side of the equation:
[tex]\[ 64n(n + 1) = 768 \][/tex]
### Step 4: Form the Second Degree Equation
Distribute and move all terms to one side of the equation to form the standard quadratic equation:
[tex]\[ 64n^2 + 64n = 768 \][/tex]
[tex]\[ 64n^2 + 64n - 768 = 0 \][/tex]
So, the quadratic equation is:
[tex]\[ 64n^2 + 64n - 768 = 0 \][/tex]
### Step 5: Solve the Quadratic Equation
To find the values of [tex]\(n\)[/tex], solve the quadratic equation using the quadratic formula or factorization method:
This quadratic equation can be solved to find the values of [tex]\(n\)[/tex]. The solutions to this quadratic equation are:
[tex]\[ n = -4 \quad \text{and} \quad n = 3 \][/tex]
### Step 6: Identify the Multiples of 8
Using the values of [tex]\(n\)[/tex], the multiples of 8 can be determined:
1. For [tex]\(n = -4\)[/tex]:
[tex]\[ 8n = 8(-4) = -32 \][/tex]
[tex]\[ 8(n + 1) = 8(-4 + 1) = 8(-3) = -24 \][/tex]
The two consecutive multiples of 8 are [tex]\(-32\)[/tex] and [tex]\(-24\)[/tex].
2. For [tex]\(n = 3\)[/tex]:
[tex]\[ 8n = 8(3) = 24 \][/tex]
[tex]\[ 8(n + 1) = 8(3 + 1) = 8(4) = 32 \][/tex]
The two consecutive multiples of 8 are [tex]\(24\)[/tex] and [tex]\(32\)[/tex].
### Step 7: Write the Conclusion
The quadratic equation formed is:
[tex]\[ 64n^2 + 64n - 768 = 0 \][/tex]
The two sets of consecutive multiples of 8 whose product is 768 are:
1. [tex]\(-32\)[/tex] and [tex]\(-24\)[/tex]
2. [tex]\(24\)[/tex] and [tex]\(32\)[/tex]
