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Answer :
Let the greater integer be [tex]$x$[/tex]. Then, the other integer is [tex]$x - 5$[/tex]. Since the product of the two integers is given as [tex]$176$[/tex], we can set up the following equation:
[tex]$$
x(x - 5) = 176.
$$[/tex]
This corresponds to option 3 in the list.
Now, to solve for [tex]$x$[/tex], we start by expanding the left-hand side:
[tex]$$
x^2 - 5x = 176.
$$[/tex]
Next, subtract [tex]$176$[/tex] from both sides to form a quadratic equation:
[tex]$$
x^2 - 5x - 176 = 0.
$$[/tex]
To solve the quadratic equation, we calculate the discriminant:
[tex]$$
\Delta = (-5)^2 - 4(1)(-176) = 25 + 704 = 729.
$$[/tex]
Since [tex]$\sqrt{729} = 27$[/tex], the quadratic formula gives:
[tex]$$
x = \frac{5 \pm 27}{2}.
$$[/tex]
This provides two possible solutions:
[tex]$$
x = \frac{5 + 27}{2} = \frac{32}{2} = 16 \quad \text{or} \quad x = \frac{5 - 27}{2} = \frac{-22}{2} = -11.
$$[/tex]
Since [tex]$x$[/tex] represents a positive integer, the valid solution is [tex]$x = 16$[/tex].
Therefore, the correct equation is
[tex]$$
x(x - 5) = 176,
$$[/tex]
which is option 3.
[tex]$$
x(x - 5) = 176.
$$[/tex]
This corresponds to option 3 in the list.
Now, to solve for [tex]$x$[/tex], we start by expanding the left-hand side:
[tex]$$
x^2 - 5x = 176.
$$[/tex]
Next, subtract [tex]$176$[/tex] from both sides to form a quadratic equation:
[tex]$$
x^2 - 5x - 176 = 0.
$$[/tex]
To solve the quadratic equation, we calculate the discriminant:
[tex]$$
\Delta = (-5)^2 - 4(1)(-176) = 25 + 704 = 729.
$$[/tex]
Since [tex]$\sqrt{729} = 27$[/tex], the quadratic formula gives:
[tex]$$
x = \frac{5 \pm 27}{2}.
$$[/tex]
This provides two possible solutions:
[tex]$$
x = \frac{5 + 27}{2} = \frac{32}{2} = 16 \quad \text{or} \quad x = \frac{5 - 27}{2} = \frac{-22}{2} = -11.
$$[/tex]
Since [tex]$x$[/tex] represents a positive integer, the valid solution is [tex]$x = 16$[/tex].
Therefore, the correct equation is
[tex]$$
x(x - 5) = 176,
$$[/tex]
which is option 3.
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