# Solve for p 67p=3p^2+84

67p=3p2+84
Subtract 3p2 from both sides of the equation.
67p-3p2=84
Move 84 to the left side of the equation by subtracting it from both sides.
67p-3p2-84=0
Factor the left side of the equation.
Let u=p. Substitute u for all occurrences of p.
67u-3u2-84
Factor by grouping.
Reorder terms.
-3u2+67u-84
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=-3⋅-84=252 and whose sum is b=67.
Factor 67 out of 67u.
-3u2+67(u)-84
Rewrite 67 as 4 plus 63
-3u2+(4+63)u-84
Apply the distributive property.
-3u2+4u+63u-84
-3u2+4u+63u-84
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
(-3u2+4u)+63u-84
Factor out the greatest common factor (GCF) from each group.
u(-3u+4)-21(-3u+4)
u(-3u+4)-21(-3u+4)
Factor the polynomial by factoring out the greatest common factor, -3u+4.
(-3u+4)(u-21)
(-3u+4)(u-21)
Replace all occurrences of u with p.
(-3p+4)(p-21)
Replace the left side with the factored expression.
(-3p+4)(p-21)=0
(-3p+4)(p-21)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
-3p+4=0
p-21=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
-3p+4=0
Subtract 4 from both sides of the equation.
-3p=-4
Divide each term by -3 and simplify.
Divide each term in -3p=-4 by -3.
-3p-3=-4-3
Cancel the common factor of -3.
Cancel the common factor.
-3p-3=-4-3
Divide p by 1.
p=-4-3
p=-4-3
Dividing two negative values results in a positive value.
p=43
p=43
p=43
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
p-21=0
Add 21 to both sides of the equation.
p=21
p=21
The final solution is all the values that make (-3p+4)(p-21)=0 true.
p=43,21
The result can be shown in multiple forms.
Exact Form:
p=43,21
Decimal Form:
p=1.3‾,21
Mixed Number Form:
p=113,21
Solve for p 67p=3p^2+84

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