Question

Solve the following equation:
9n² - n - 2/3 = 0

Answer 1

see explanation

Explanation:

Given

9n² - n -
`(2)/(3)` = 0 ← multiply through by 3 to clear the fraction

27n² - 3n - 2 = 0 ← factoring

Consider the factors of the product of the n² term and the constant term which sum to give the coefficient of the n- term

product = 27 × - 2 = - 54 and sum = - 3

The factors are - 9 and + 6

Use these factors to split the n- term

27n² - 9n + 6n - 2 = 0 ( factor the first/second and third/fourth terms )

9n(3n - 1) + 2(3n - 1) = 0 ← factor out (3n - 1) from each term

(3n - 1)(9n + 2) = 0

Equate each factor to zero and solve for n

3n - 1 = 0 ⇒ 3n = 1 ⇒ n =
`(1)/(3)`

9n + 2 = 0 ⇒ 9n = - 2 ⇒ n = -
`(2)/(9)`