Alright! Let’s walk through the solution step-by-step to determine which equation can be factored and solved for the smaller integer when the product of two consecutive integers is 72.
1. Understanding the Problem:
- The problem states we have two consecutive integers whose product is 72.
- Let's denote these two consecutive integers by [tex]\( x \)[/tex] and [tex]\( x+1 \)[/tex] where [tex]\( x \)[/tex] is the smaller integer.
2. Setting Up the Equation:
- Given the consecutive integers are [tex]\( x \)[/tex] and [tex]\( x+1 \)[/tex], their product can be expressed by the equation:
[tex]\[
x(x + 1) = 72
\][/tex]
3. Forming a Quadratic Equation:
- Expanding this equation:
[tex]\[
x(x + 1) = x^2 + x
\][/tex]
- Thus, the equation becomes:
[tex]\[
x^2 + x = 72
\][/tex]
- To form a standard quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], we move 72 to the left side of the equation:
[tex]\[
x^2 + x - 72 = 0
\][/tex]
4. Comparing with Given Choices:
- The equation we derived is [tex]\( x^2 + x - 72 = 0 \)[/tex].
- Checking the given options:
- [tex]\( x^2 + x - 72 = 0 \)[/tex]
- [tex]\( x^2 + x + 72 = 0 \)[/tex]
- [tex]\( x^2 + 2x - 72 = 0 \)[/tex]
- [tex]\( x^2 + 2x + 72 = 0 \)[/tex]
The correct equation that we have derived and matches one of the given choices is:
[tex]\[
x^2 + x - 72 = 0
\][/tex]
5. Solving the Quadratic Equation:
- To find the smaller integer [tex]\( x \)[/tex], solve the quadratic equation [tex]\( x^2 + x - 72 = 0 \)[/tex].
- The solutions to this equation are [tex]\( x = -9 \)[/tex] and [tex]\( x = 8 \)[/tex].
Therefore, the equation that can be factored and solved for the smaller integer is:
[tex]\[
x^2 + x - 72 = 0
\][/tex]
