24n^2-40n-54=0

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Solution for 24n^2-40n-54=0 equation:


Simplifying
24n2 + -40n + -54 = 0

Reorder the terms:
-54 + -40n + 24n2 = 0

Solving
-54 + -40n + 24n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-27 + -20n + 12n2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(-27 + -20n + 12n2)' equal to zero and attempt to solve: Simplifying -27 + -20n + 12n2 = 0 Solving -27 + -20n + 12n2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -2.25 + -1.666666667n + n2 = 0 Move the constant term to the right: Add '2.25' to each side of the equation. -2.25 + -1.666666667n + 2.25 + n2 = 0 + 2.25 Reorder the terms: -2.25 + 2.25 + -1.666666667n + n2 = 0 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + -1.666666667n + n2 = 0 + 2.25 -1.666666667n + n2 = 0 + 2.25 Combine like terms: 0 + 2.25 = 2.25 -1.666666667n + n2 = 2.25 The n term is -1.666666667n. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667n + 0.6944444447 + n2 = 2.25 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667n + n2 = 2.25 + 0.6944444447 Combine like terms: 2.25 + 0.6944444447 = 2.9444444447 0.6944444447 + -1.666666667n + n2 = 2.9444444447 Factor a perfect square on the left side: (n + -0.8333333335)(n + -0.8333333335) = 2.9444444447 Calculate the square root of the right side: 1.715938357 Break this problem into two subproblems by setting (n + -0.8333333335) equal to 1.715938357 and -1.715938357.

Subproblem 1

n + -0.8333333335 = 1.715938357 Simplifying n + -0.8333333335 = 1.715938357 Reorder the terms: -0.8333333335 + n = 1.715938357 Solving -0.8333333335 + n = 1.715938357 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + n = 1.715938357 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + n = 1.715938357 + 0.8333333335 n = 1.715938357 + 0.8333333335 Combine like terms: 1.715938357 + 0.8333333335 = 2.5492716905 n = 2.5492716905 Simplifying n = 2.5492716905

Subproblem 2

n + -0.8333333335 = -1.715938357 Simplifying n + -0.8333333335 = -1.715938357 Reorder the terms: -0.8333333335 + n = -1.715938357 Solving -0.8333333335 + n = -1.715938357 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + n = -1.715938357 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + n = -1.715938357 + 0.8333333335 n = -1.715938357 + 0.8333333335 Combine like terms: -1.715938357 + 0.8333333335 = -0.8826050235 n = -0.8826050235 Simplifying n = -0.8826050235

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.5492716905, -0.8826050235}

Solution

n = {2.5492716905, -0.8826050235}

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