5n^2+90=-90n

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Solution for 5n^2+90=-90n equation:


Simplifying
5n2 + 90 = -90n

Reorder the terms:
90 + 5n2 = -90n

Solving
90 + 5n2 = -90n

Solving for variable 'n'.

Reorder the terms:
90 + 90n + 5n2 = -90n + 90n

Combine like terms: -90n + 90n = 0
90 + 90n + 5n2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(18 + 18n + n2) = 0

Ignore the factor 5.

Subproblem 1

Set the factor '(18 + 18n + n2)' equal to zero and attempt to solve: Simplifying 18 + 18n + n2 = 0 Solving 18 + 18n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '-18' to each side of the equation. 18 + 18n + -18 + n2 = 0 + -18 Reorder the terms: 18 + -18 + 18n + n2 = 0 + -18 Combine like terms: 18 + -18 = 0 0 + 18n + n2 = 0 + -18 18n + n2 = 0 + -18 Combine like terms: 0 + -18 = -18 18n + n2 = -18 The n term is 18n. Take half its coefficient (9). Square it (81) and add it to both sides. Add '81' to each side of the equation. 18n + 81 + n2 = -18 + 81 Reorder the terms: 81 + 18n + n2 = -18 + 81 Combine like terms: -18 + 81 = 63 81 + 18n + n2 = 63 Factor a perfect square on the left side: (n + 9)(n + 9) = 63 Calculate the square root of the right side: 7.937253933 Break this problem into two subproblems by setting (n + 9) equal to 7.937253933 and -7.937253933.

Subproblem 1

n + 9 = 7.937253933 Simplifying n + 9 = 7.937253933 Reorder the terms: 9 + n = 7.937253933 Solving 9 + n = 7.937253933 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + n = 7.937253933 + -9 Combine like terms: 9 + -9 = 0 0 + n = 7.937253933 + -9 n = 7.937253933 + -9 Combine like terms: 7.937253933 + -9 = -1.062746067 n = -1.062746067 Simplifying n = -1.062746067

Subproblem 2

n + 9 = -7.937253933 Simplifying n + 9 = -7.937253933 Reorder the terms: 9 + n = -7.937253933 Solving 9 + n = -7.937253933 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + n = -7.937253933 + -9 Combine like terms: 9 + -9 = 0 0 + n = -7.937253933 + -9 n = -7.937253933 + -9 Combine like terms: -7.937253933 + -9 = -16.937253933 n = -16.937253933 Simplifying n = -16.937253933

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-1.062746067, -16.937253933}

Solution

n = {-1.062746067, -16.937253933}

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