4n^2+9n-3=0

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Solution for 4n^2+9n-3=0 equation:


Simplifying
4n2 + 9n + -3 = 0

Reorder the terms:
-3 + 9n + 4n2 = 0

Solving
-3 + 9n + 4n2 = 0

Solving for variable 'n'.

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

Divide each side by '4'.
-0.75 + 2.25n + n2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + 2.25n + 0.75 + n2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + 2.25n + n2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + 2.25n + n2 = 0 + 0.75
2.25n + n2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
2.25n + n2 = 0.75

The n term is 2.25n.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25n + 1.265625 + n2 = 0.75 + 1.265625

Reorder the terms:
1.265625 + 2.25n + n2 = 0.75 + 1.265625

Combine like terms: 0.75 + 1.265625 = 2.015625
1.265625 + 2.25n + n2 = 2.015625

Factor a perfect square on the left side:
(n + 1.125)(n + 1.125) = 2.015625

Calculate the square root of the right side: 1.419727086

Break this problem into two subproblems by setting 
(n + 1.125) equal to 1.419727086 and -1.419727086.

Subproblem 1

n + 1.125 = 1.419727086 Simplifying n + 1.125 = 1.419727086 Reorder the terms: 1.125 + n = 1.419727086 Solving 1.125 + n = 1.419727086 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + n = 1.419727086 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + n = 1.419727086 + -1.125 n = 1.419727086 + -1.125 Combine like terms: 1.419727086 + -1.125 = 0.294727086 n = 0.294727086 Simplifying n = 0.294727086

Subproblem 2

n + 1.125 = -1.419727086 Simplifying n + 1.125 = -1.419727086 Reorder the terms: 1.125 + n = -1.419727086 Solving 1.125 + n = -1.419727086 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + n = -1.419727086 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + n = -1.419727086 + -1.125 n = -1.419727086 + -1.125 Combine like terms: -1.419727086 + -1.125 = -2.544727086 n = -2.544727086 Simplifying n = -2.544727086

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.294727086, -2.544727086}

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