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

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.

-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.

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