help on quadratic

Question

12m^3+2=3m+8m^2

Answer 1

12m^3 - 8m^2 -3m + 2 = 0

notice 12/8 = 3/2, the coefficients of the 1st two terms are in the same ratio as the last two terms. factor out 4m^2 from the 1st two terms

= 4m^2(3m-2) -3m+2 = 0

factor out -1 from the last 2 terms

= 4m^2(3m-2) -(3m-2) = 0

= (4m^2 -1)(3m-2) = 0

set each factor = 0 and solve for m

4m^2=1

m^2 = 1/4

m= + or - 1/2

3m-2 = 0

3m=2

m= 2/3

m = -1/2, 1/2, 2/3

Answer 2

STEP 1:

set all terms to one side, such that the other side equals zero

12m³ + 2 -3m -8m² = 3m -3m + 8m² -8m²

12m³ - 8m2 - 3m + 2 = 0 = 3m -3m + 8m² -8m²

STEP 2:

since there are four unique terms in descending order, try Factor by Grouping

4m²(3m - 2) + -3m + 2 = 0

notice, the last two terms are almost the same as the binomial terms, only their signs are opposite

4m²(3m - 2) + -3m + 2 = 0

factor out a negative from each of the last two terms

4m²(3m - 2) + -( 3m -2) = 0

4m²(3m - 2) - (3m - 2) = 0

notice, each of the two terms now share a common factor of (3m - 2)

factor it out

(3m - 2)(4m² - 1) = 0

notice, the second binomial is a Difference of Squares, which can be factored further

(3m - 2)(2m + 1)(2m - 1) = 0

STEP 3:

Solve for m by using the Zero Product Property

3m - 2 = 0; m = 2/3

2m + 1 = 0; m = -1/2

2m - 1 = 0; m = 1/2

STEP CHECK:

check your work by plugging each solution into the original equation to confirm it is in fact a true statement

for m = 2/3

12(2/3)³ + 2 = 3(2/3) + 8(2/3)²

50/9 = 50/9 √

for m = -1/2

12(-1/2)³ + 2 = 3(-1/2) + 8(-1/2)²

1/2 = 1/2 √

for m = 1/2

12(1/2)³ + 2 = 3(1/2) + 8(1/2)²

7/2 = 7/2 √

ANSWER:

m = 2/3, -1/2, 1/2