Answer :
We are given two equations and asked which one has both [tex]$-7$[/tex] and [tex]$7$[/tex] as solutions for [tex]$f$[/tex]. Let's analyze both:
1. For the equation
[tex]$$f^2 = 49,$$[/tex]
we can take the square root of both sides. Since
[tex]$$\sqrt{49} = 7,$$[/tex]
and the square of both [tex]$7$[/tex] and [tex]$-7$[/tex] is [tex]$49$[/tex], the solutions are
[tex]$$f = 7 \quad \text{or} \quad f = -7.$$[/tex]
This equation has both [tex]$-7$[/tex] and [tex]$7$[/tex] as valid solutions.
2. For the equation
[tex]$$f^3 = 343,$$[/tex]
we take the cube root of both sides. The cube root of [tex]$343$[/tex] is:
[tex]$$\sqrt[3]{343} = 7.$$[/tex]
Since [tex]$(-7)^3 = -343 \neq 343$[/tex], the equation does not have [tex]$-7$[/tex] as a solution.
Thus, the only equation that has both [tex]$-7$[/tex] and [tex]$7$[/tex] as solutions is
[tex]$$f^2 = 49.$$[/tex]
Therefore, the answer is option 4.
1. For the equation
[tex]$$f^2 = 49,$$[/tex]
we can take the square root of both sides. Since
[tex]$$\sqrt{49} = 7,$$[/tex]
and the square of both [tex]$7$[/tex] and [tex]$-7$[/tex] is [tex]$49$[/tex], the solutions are
[tex]$$f = 7 \quad \text{or} \quad f = -7.$$[/tex]
This equation has both [tex]$-7$[/tex] and [tex]$7$[/tex] as valid solutions.
2. For the equation
[tex]$$f^3 = 343,$$[/tex]
we take the cube root of both sides. The cube root of [tex]$343$[/tex] is:
[tex]$$\sqrt[3]{343} = 7.$$[/tex]
Since [tex]$(-7)^3 = -343 \neq 343$[/tex], the equation does not have [tex]$-7$[/tex] as a solution.
Thus, the only equation that has both [tex]$-7$[/tex] and [tex]$7$[/tex] as solutions is
[tex]$$f^2 = 49.$$[/tex]
Therefore, the answer is option 4.
