Answer :
We start with the equation
[tex]$$
-3f^2 - 84f = -3.
$$[/tex]
Step 1. Bring all terms to one side.
Add 3 to both sides:
[tex]$$
-3f^2 - 84f + 3 = 0.
$$[/tex]
Multiply the entire equation by [tex]$-1$[/tex] to simplify the signs:
[tex]$$
3f^2 + 84f - 3 = 0.
$$[/tex]
Step 2. Divide by 3.
Divide every term by [tex]$3$[/tex] to obtain:
[tex]$$
f^2 + 28f - 1 = 0.
$$[/tex]
Step 3. Complete the square.
Focus on the quadratic and linear terms on the left:
[tex]$$
f^2 + 28f.
$$[/tex]
Take half of the coefficient of [tex]$f$[/tex], which is [tex]$14$[/tex], and square it to get [tex]$196$[/tex]. Add and subtract [tex]$196$[/tex]:
[tex]$$
f^2 + 28f = \left(f^2 + 28f + 196\right) - 196 = (f+14)^2 - 196.
$$[/tex]
Substitute this back into the equation:
[tex]$$
(f+14)^2 - 196 - 1 = 0.
$$[/tex]
Simplify by combining the constant terms:
[tex]$$
(f+14)^2 - 197 = 0.
$$[/tex]
So,
[tex]$$
(f+14)^2 = 197.
$$[/tex]
Step 4. Solve for [tex]$f$[/tex].
Take the square root of both sides:
[tex]$$
f + 14 = \pm \sqrt{197}.
$$[/tex]
Subtract [tex]$14$[/tex] from both sides to isolate [tex]$f$[/tex]:
[tex]$$
f = -14 \pm \sqrt{197}.
$$[/tex]
Thus, the solutions are
[tex]$$
f = -14 + \sqrt{197} \quad \text{or} \quad f = -14 - \sqrt{197}.
$$[/tex]
Step 5. Approximating the values.
Given that
[tex]$$
\sqrt{197} \approx 14.035668847618199,
$$[/tex]
we find:
1. For [tex]$f = -14 + \sqrt{197}$[/tex]:
[tex]$$
f \approx -14 + 14.035668847618199 \approx 0.035668847618198996 \approx 0.04 \quad (\text{rounded to the nearest hundredth}).
$$[/tex]
2. For [tex]$f = -14 - \sqrt{197}$[/tex]:
[tex]$$
f \approx -14 - 14.035668847618199 \approx -28.035668847618197 \approx -28.04 \quad (\text{rounded to the nearest hundredth}).
$$[/tex]
Final Answer:
[tex]$$
f = -14 + \sqrt{197} \quad \text{or} \quad f = -14 - \sqrt{197},
$$[/tex]
which approximates to
[tex]$$
f \approx 0.04 \quad \text{or} \quad f \approx -28.04.
$$[/tex]
[tex]$$
-3f^2 - 84f = -3.
$$[/tex]
Step 1. Bring all terms to one side.
Add 3 to both sides:
[tex]$$
-3f^2 - 84f + 3 = 0.
$$[/tex]
Multiply the entire equation by [tex]$-1$[/tex] to simplify the signs:
[tex]$$
3f^2 + 84f - 3 = 0.
$$[/tex]
Step 2. Divide by 3.
Divide every term by [tex]$3$[/tex] to obtain:
[tex]$$
f^2 + 28f - 1 = 0.
$$[/tex]
Step 3. Complete the square.
Focus on the quadratic and linear terms on the left:
[tex]$$
f^2 + 28f.
$$[/tex]
Take half of the coefficient of [tex]$f$[/tex], which is [tex]$14$[/tex], and square it to get [tex]$196$[/tex]. Add and subtract [tex]$196$[/tex]:
[tex]$$
f^2 + 28f = \left(f^2 + 28f + 196\right) - 196 = (f+14)^2 - 196.
$$[/tex]
Substitute this back into the equation:
[tex]$$
(f+14)^2 - 196 - 1 = 0.
$$[/tex]
Simplify by combining the constant terms:
[tex]$$
(f+14)^2 - 197 = 0.
$$[/tex]
So,
[tex]$$
(f+14)^2 = 197.
$$[/tex]
Step 4. Solve for [tex]$f$[/tex].
Take the square root of both sides:
[tex]$$
f + 14 = \pm \sqrt{197}.
$$[/tex]
Subtract [tex]$14$[/tex] from both sides to isolate [tex]$f$[/tex]:
[tex]$$
f = -14 \pm \sqrt{197}.
$$[/tex]
Thus, the solutions are
[tex]$$
f = -14 + \sqrt{197} \quad \text{or} \quad f = -14 - \sqrt{197}.
$$[/tex]
Step 5. Approximating the values.
Given that
[tex]$$
\sqrt{197} \approx 14.035668847618199,
$$[/tex]
we find:
1. For [tex]$f = -14 + \sqrt{197}$[/tex]:
[tex]$$
f \approx -14 + 14.035668847618199 \approx 0.035668847618198996 \approx 0.04 \quad (\text{rounded to the nearest hundredth}).
$$[/tex]
2. For [tex]$f = -14 - \sqrt{197}$[/tex]:
[tex]$$
f \approx -14 - 14.035668847618199 \approx -28.035668847618197 \approx -28.04 \quad (\text{rounded to the nearest hundredth}).
$$[/tex]
Final Answer:
[tex]$$
f = -14 + \sqrt{197} \quad \text{or} \quad f = -14 - \sqrt{197},
$$[/tex]
which approximates to
[tex]$$
f \approx 0.04 \quad \text{or} \quad f \approx -28.04.
$$[/tex]
