# Solve for a f(3a)=2(3a)^2-3(3a)+4

f(3a)=2(3a)2-3(3a)+4
Since a is on the right side of the equation, switch the sides so it is on the left side of the equation.
2(3a)2-3(3a)+4=f(3a)
Simplify each term.
Apply the product rule to 3a.
2(32a2)-3(3a)+4=f(3a)
Raise 3 to the power of 2.
2(9a2)-3(3a)+4=f(3a)
Multiply 9 by 2.
18a2-3(3a)+4=f(3a)
Multiply 3 by -3.
18a2-9a+4=f(3a)
18a2-9a+4=f(3a)
Rewrite using the commutative property of multiplication.
18a2-9a+4=3fa
Subtract 3fa from both sides of the equation.
18a2-9a+4-3fa=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=18, b=-9-3f, and c=4 into the quadratic formula and solve for a.
-(-9-3f)±(-9-3f)2-4⋅(18⋅4)2⋅18
Simplify.
Simplify the numerator.
Apply the distributive property.
a=9-(-3f)±(-9-3f)2-4⋅(18⋅4)2⋅18
Multiply -1 by -9.
a=9-(-3f)±(-9-3f)2-4⋅(18⋅4)2⋅18
Multiply -3 by -1.
a=9+3f±(-9-3f)2-4⋅(18⋅4)2⋅18
Rewrite (-9-3f)2 as (-9-3f)(-9-3f).
a=9+3f±(-9-3f)(-9-3f)-4⋅(18⋅4)2⋅18
Expand (-9-3f)(-9-3f) using the FOIL Method.
Apply the distributive property.
a=9+3f±-9(-9-3f)-3f(-9-3f)-4⋅(18⋅4)2⋅18
Apply the distributive property.
a=9+3f±-9⋅-9-9(-3f)-3f(-9-3f)-4⋅(18⋅4)2⋅18
Apply the distributive property.
a=9+3f±-9⋅-9-9(-3f)-3f⋅-9-3f(-3f)-4⋅(18⋅4)2⋅18
a=9+3f±-9⋅-9-9(-3f)-3f⋅-9-3f(-3f)-4⋅(18⋅4)2⋅18
Simplify and combine like terms.
Simplify each term.
Multiply -9 by -9.
a=9+3f±81-9(-3f)-3f⋅-9-3f(-3f)-4⋅(18⋅4)2⋅18
Multiply -3 by -9.
a=9+3f±81+27f-3f⋅-9-3f(-3f)-4⋅(18⋅4)2⋅18
Multiply -9 by -3.
a=9+3f±81+27f+27f-3f(-3f)-4⋅(18⋅4)2⋅18
Multiply f by f.
a=9+3f±81+27f+27f-3⋅(-3f2)-4⋅(18⋅4)2⋅18
Multiply -3 by -3.
a=9+3f±81+27f+27f+9f2-4⋅(18⋅4)2⋅18
a=9+3f±81+27f+27f+9f2-4⋅(18⋅4)2⋅18
a=9+3f±81+54f+9f2-4⋅(18⋅4)2⋅18
a=9+3f±81+54f+9f2-4⋅(18⋅4)2⋅18
Multiply 18 by 4.
a=9+3f±81+54f+9f2-4⋅722⋅18
Multiply -4 by 72.
a=9+3f±81+54f+9f2-2882⋅18
Subtract 288 from 81.
a=9+3f±54f+9f2-2072⋅18
Reorder terms.
a=9+3f±9f2+54f-2072⋅18
Factor 9 out of 9f2+54f-207.
Factor 9 out of 9f2.
a=9+3f±9(f2)+54f-2072⋅18
Factor 9 out of 54f.
a=9+3f±9(f2)+9(6f)-2072⋅18
Factor 9 out of -207.
a=9+3f±9f2+9(6f)+9⋅-232⋅18
Factor 9 out of 9f2+9(6f).
a=9+3f±9(f2+6f)+9⋅-232⋅18
Factor 9 out of 9(f2+6f)+9⋅-23.
a=9+3f±9(f2+6f-23)2⋅18
a=9+3f±9(f2+6f-23)2⋅18
Rewrite 9 as 32.
a=9+3f±32(f2+6f-23)2⋅18
Pull terms out from under the radical.
a=9+3f±3f2+6f-232⋅18
a=9+3f±3f2+6f-232⋅18
Multiply 2 by 18.
a=9+3f±3f2+6f-2336
a=9+3f±3f2+6f-2336
The final answer is the combination of both solutions.
a=3+f+f2+6f-2312
a=3+f-f2+6f-2312
Solve for a f(3a)=2(3a)^2-3(3a)+4

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