n^2-n-21=72

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Solution for n^2-n-21=72 equation:


Simplifying
n2 + -1n + -21 = 72

Reorder the terms:
-21 + -1n + n2 = 72

Solving
-21 + -1n + n2 = 72

Solving for variable 'n'.

Reorder the terms:
-21 + -72 + -1n + n2 = 72 + -72

Combine like terms: -21 + -72 = -93
-93 + -1n + n2 = 72 + -72

Combine like terms: 72 + -72 = 0
-93 + -1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '93' to each side of the equation.
-93 + -1n + 93 + n2 = 0 + 93

Reorder the terms:
-93 + 93 + -1n + n2 = 0 + 93

Combine like terms: -93 + 93 = 0
0 + -1n + n2 = 0 + 93
-1n + n2 = 0 + 93

Combine like terms: 0 + 93 = 93
-1n + n2 = 93

The n term is -1n.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1n + 0.25 + n2 = 93 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 93 + 0.25

Combine like terms: 93 + 0.25 = 93.25
0.25 + -1n + n2 = 93.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 93.25

Calculate the square root of the right side: 9.656603958

Break this problem into two subproblems by setting 
(n + -0.5) equal to 9.656603958 and -9.656603958.

Subproblem 1

n + -0.5 = 9.656603958 Simplifying n + -0.5 = 9.656603958 Reorder the terms: -0.5 + n = 9.656603958 Solving -0.5 + n = 9.656603958 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 9.656603958 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 9.656603958 + 0.5 n = 9.656603958 + 0.5 Combine like terms: 9.656603958 + 0.5 = 10.156603958 n = 10.156603958 Simplifying n = 10.156603958

Subproblem 2

n + -0.5 = -9.656603958 Simplifying n + -0.5 = -9.656603958 Reorder the terms: -0.5 + n = -9.656603958 Solving -0.5 + n = -9.656603958 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -9.656603958 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -9.656603958 + 0.5 n = -9.656603958 + 0.5 Combine like terms: -9.656603958 + 0.5 = -9.156603958 n = -9.156603958 Simplifying n = -9.156603958

Solution

The solution to the problem is based on the solutions from the subproblems. n = {10.156603958, -9.156603958}

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