Completing the rectangular simply manner taking the primary two terms and making a ideal rectangular out of it. Consider concerning the pattern of the perfect square: (x + a)^2 = x^2 + 2ax + a^2 (x - a)^2 = x^2 - 2ax + a^2 First off, you'll be able to detect that the sign of the linear term tells you which one you are coping with. If the center term is negative, it's (x - a)^2. If positive, it's (x + a)^2. Additionally, notice that in both instances, the coefficient of the linear term is twice the rectangular root of the constant term. So: 1) n^2 + 16n - 7 = 0 We're coping with an (x + a)^2 drawback here, considering the fact that the 16n is constructive. So, from the pattern, n^2 + 2an + a^2 = zero n^2 + 16n + a^2 = 0 certainly, 2a is sixteen, so a = eight Now we need to therapeutic massage the equation into the right type. The ultimate rectangular can be n^2 + 16n + 64, but we've n^2 + 16n - 7. What can we must add to the equation to show -7 into +sixty four? Evidently, it is 71. So: n^2 + 16n - 7 = zero n^2 + 16n - 7 + 71 = zero + seventy one n^2 + 16n + 64 = 71 (n + 8)^2 = 71 Now we can resolve it: n + eight = +/- sqrt(71) (sqrt() = rectangular root) n = -8 +/- sqrt(71) 2) For this one, we have to get difficult. Each of the patterns involve a unary quadratic term - i.E. The coefficient of the squared time period must be 1. So, we by way of doing that: 3x^2 - 5x + 2 = zero x^2 - 5/three x + 2/3 = 0 Now: take 5/3 and cut it in half to get 5/6. Square that and get 25/36. So the superb square we want is x^2 - 5/3 x + 25/36 = (x - 5/6)^2 We ought to add anything to turn 2/3 into 25/36. 2/three = 24/36, so we have got to add 1/36. X^2 - 5/3x + 2/three + 1/36 = 1/36 x^2 - 5/3 x + 25/36 = 1/36 (x - 5/6)^2 = 1/36 x - 5/6 = +/- 1/6 x = 5/6 +/- 1/6 So, x = 1 or x = 4/6 = 2/3 Work the 1/3 one the equal approach as #2.