5n^2+7n-51=0

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Solution for 5n^2+7n-51=0 equation:


Simplifying
5n2 + 7n + -51 = 0

Reorder the terms:
-51 + 7n + 5n2 = 0

Solving
-51 + 7n + 5n2 = 0

Solving for variable 'n'.

Begin completing the square.  Divide all terms by
5 the coefficient of the squared term: 

Divide each side by '5'.
-10.2 + 1.4n + n2 = 0

Move the constant term to the right:

Add '10.2' to each side of the equation.
-10.2 + 1.4n + 10.2 + n2 = 0 + 10.2

Reorder the terms:
-10.2 + 10.2 + 1.4n + n2 = 0 + 10.2

Combine like terms: -10.2 + 10.2 = 0.0
0.0 + 1.4n + n2 = 0 + 10.2
1.4n + n2 = 0 + 10.2

Combine like terms: 0 + 10.2 = 10.2
1.4n + n2 = 10.2

The n term is 1.4n.  Take half its coefficient (0.7).
Square it (0.49) and add it to both sides.

Add '0.49' to each side of the equation.
1.4n + 0.49 + n2 = 10.2 + 0.49

Reorder the terms:
0.49 + 1.4n + n2 = 10.2 + 0.49

Combine like terms: 10.2 + 0.49 = 10.69
0.49 + 1.4n + n2 = 10.69

Factor a perfect square on the left side:
(n + 0.7)(n + 0.7) = 10.69

Calculate the square root of the right side: 3.269556545

Break this problem into two subproblems by setting 
(n + 0.7) equal to 3.269556545 and -3.269556545.

Subproblem 1

n + 0.7 = 3.269556545 Simplifying n + 0.7 = 3.269556545 Reorder the terms: 0.7 + n = 3.269556545 Solving 0.7 + n = 3.269556545 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + n = 3.269556545 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + n = 3.269556545 + -0.7 n = 3.269556545 + -0.7 Combine like terms: 3.269556545 + -0.7 = 2.569556545 n = 2.569556545 Simplifying n = 2.569556545

Subproblem 2

n + 0.7 = -3.269556545 Simplifying n + 0.7 = -3.269556545 Reorder the terms: 0.7 + n = -3.269556545 Solving 0.7 + n = -3.269556545 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + n = -3.269556545 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + n = -3.269556545 + -0.7 n = -3.269556545 + -0.7 Combine like terms: -3.269556545 + -0.7 = -3.969556545 n = -3.969556545 Simplifying n = -3.969556545

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.569556545, -3.969556545}

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