4n^2+7n-5=0

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Solution for 4n^2+7n-5=0 equation:


Simplifying
4n2 + 7n + -5 = 0

Reorder the terms:
-5 + 7n + 4n2 = 0

Solving
-5 + 7n + 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'.
-1.25 + 1.75n + n2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + 1.75n + 1.25 + n2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + 1.75n + n2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + 1.75n + n2 = 0 + 1.25
1.75n + n2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
1.75n + n2 = 1.25

The n term is 1.75n.  Take half its coefficient (0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
1.75n + 0.765625 + n2 = 1.25 + 0.765625

Reorder the terms:
0.765625 + 1.75n + n2 = 1.25 + 0.765625

Combine like terms: 1.25 + 0.765625 = 2.015625
0.765625 + 1.75n + n2 = 2.015625

Factor a perfect square on the left side:
(n + 0.875)(n + 0.875) = 2.015625

Calculate the square root of the right side: 1.419727086

Break this problem into two subproblems by setting 
(n + 0.875) equal to 1.419727086 and -1.419727086.

Subproblem 1

n + 0.875 = 1.419727086 Simplifying n + 0.875 = 1.419727086 Reorder the terms: 0.875 + n = 1.419727086 Solving 0.875 + n = 1.419727086 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + n = 1.419727086 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = 1.419727086 + -0.875 n = 1.419727086 + -0.875 Combine like terms: 1.419727086 + -0.875 = 0.544727086 n = 0.544727086 Simplifying n = 0.544727086

Subproblem 2

n + 0.875 = -1.419727086 Simplifying n + 0.875 = -1.419727086 Reorder the terms: 0.875 + n = -1.419727086 Solving 0.875 + n = -1.419727086 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + n = -1.419727086 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = -1.419727086 + -0.875 n = -1.419727086 + -0.875 Combine like terms: -1.419727086 + -0.875 = -2.294727086 n = -2.294727086 Simplifying n = -2.294727086

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.544727086, -2.294727086}

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