4n^2+6n-16=-15n^2

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Solution for 4n^2+6n-16=-15n^2 equation:


Simplifying
4n2 + 6n + -16 = -15n2

Reorder the terms:
-16 + 6n + 4n2 = -15n2

Solving
-16 + 6n + 4n2 = -15n2

Solving for variable 'n'.

Combine like terms: 4n2 + 15n2 = 19n2
-16 + 6n + 19n2 = -15n2 + 15n2

Combine like terms: -15n2 + 15n2 = 0
-16 + 6n + 19n2 = 0

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

Divide each side by '19'.
-0.8421052632 + 0.3157894737n + n2 = 0

Move the constant term to the right:

Add '0.8421052632' to each side of the equation.
-0.8421052632 + 0.3157894737n + 0.8421052632 + n2 = 0 + 0.8421052632

Reorder the terms:
-0.8421052632 + 0.8421052632 + 0.3157894737n + n2 = 0 + 0.8421052632

Combine like terms: -0.8421052632 + 0.8421052632 = 0.0000000000
0.0000000000 + 0.3157894737n + n2 = 0 + 0.8421052632
0.3157894737n + n2 = 0 + 0.8421052632

Combine like terms: 0 + 0.8421052632 = 0.8421052632
0.3157894737n + n2 = 0.8421052632

The n term is 0.3157894737n.  Take half its coefficient (0.1578947369).
Square it (0.02493074794) and add it to both sides.

Add '0.02493074794' to each side of the equation.
0.3157894737n + 0.02493074794 + n2 = 0.8421052632 + 0.02493074794

Reorder the terms:
0.02493074794 + 0.3157894737n + n2 = 0.8421052632 + 0.02493074794

Combine like terms: 0.8421052632 + 0.02493074794 = 0.86703601114
0.02493074794 + 0.3157894737n + n2 = 0.86703601114

Factor a perfect square on the left side:
(n + 0.1578947369)(n + 0.1578947369) = 0.86703601114

Calculate the square root of the right side: 0.931147685

Break this problem into two subproblems by setting 
(n + 0.1578947369) equal to 0.931147685 and -0.931147685.

Subproblem 1

n + 0.1578947369 = 0.931147685 Simplifying n + 0.1578947369 = 0.931147685 Reorder the terms: 0.1578947369 + n = 0.931147685 Solving 0.1578947369 + n = 0.931147685 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1578947369' to each side of the equation. 0.1578947369 + -0.1578947369 + n = 0.931147685 + -0.1578947369 Combine like terms: 0.1578947369 + -0.1578947369 = 0.0000000000 0.0000000000 + n = 0.931147685 + -0.1578947369 n = 0.931147685 + -0.1578947369 Combine like terms: 0.931147685 + -0.1578947369 = 0.7732529481 n = 0.7732529481 Simplifying n = 0.7732529481

Subproblem 2

n + 0.1578947369 = -0.931147685 Simplifying n + 0.1578947369 = -0.931147685 Reorder the terms: 0.1578947369 + n = -0.931147685 Solving 0.1578947369 + n = -0.931147685 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1578947369' to each side of the equation. 0.1578947369 + -0.1578947369 + n = -0.931147685 + -0.1578947369 Combine like terms: 0.1578947369 + -0.1578947369 = 0.0000000000 0.0000000000 + n = -0.931147685 + -0.1578947369 n = -0.931147685 + -0.1578947369 Combine like terms: -0.931147685 + -0.1578947369 = -1.0890424219 n = -1.0890424219 Simplifying n = -1.0890424219

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.7732529481, -1.0890424219}

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