4n^2-6n-7=0

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Solution for 4n^2-6n-7=0 equation:


Simplifying
4n2 + -6n + -7 = 0

Reorder the terms:
-7 + -6n + 4n2 = 0

Solving
-7 + -6n + 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'.
-1.75 + -1.5n + n2 = 0

Move the constant term to the right:

Add '1.75' to each side of the equation.
-1.75 + -1.5n + 1.75 + n2 = 0 + 1.75

Reorder the terms:
-1.75 + 1.75 + -1.5n + n2 = 0 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + -1.5n + n2 = 0 + 1.75
-1.5n + n2 = 0 + 1.75

Combine like terms: 0 + 1.75 = 1.75
-1.5n + n2 = 1.75

The n term is -1.5n.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5n + 0.5625 + n2 = 1.75 + 0.5625

Reorder the terms:
0.5625 + -1.5n + n2 = 1.75 + 0.5625

Combine like terms: 1.75 + 0.5625 = 2.3125
0.5625 + -1.5n + n2 = 2.3125

Factor a perfect square on the left side:
(n + -0.75)(n + -0.75) = 2.3125

Calculate the square root of the right side: 1.520690633

Break this problem into two subproblems by setting 
(n + -0.75) equal to 1.520690633 and -1.520690633.

Subproblem 1

n + -0.75 = 1.520690633 Simplifying n + -0.75 = 1.520690633 Reorder the terms: -0.75 + n = 1.520690633 Solving -0.75 + n = 1.520690633 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 1.520690633 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 1.520690633 + 0.75 n = 1.520690633 + 0.75 Combine like terms: 1.520690633 + 0.75 = 2.270690633 n = 2.270690633 Simplifying n = 2.270690633

Subproblem 2

n + -0.75 = -1.520690633 Simplifying n + -0.75 = -1.520690633 Reorder the terms: -0.75 + n = -1.520690633 Solving -0.75 + n = -1.520690633 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -1.520690633 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -1.520690633 + 0.75 n = -1.520690633 + 0.75 Combine like terms: -1.520690633 + 0.75 = -0.770690633 n = -0.770690633 Simplifying n = -0.770690633

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.270690633, -0.770690633}

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