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Simplifying 4n(5n2 + -7n + -3) = 0 Reorder the terms: 4n(-3 + -7n + 5n2) = 0 (-3 * 4n + -7n * 4n + 5n2 * 4n) = 0 (-12n + -28n2 + 20n3) = 0 Solving -12n + -28n2 + 20n3 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '4n'. 4n(-3 + -7n + 5n2) = 0 Ignore the factor 4.Subproblem 1
Set the factor 'n' equal to zero and attempt to solve: Simplifying n = 0 Solving n = 0 Move all terms containing n to the left, all other terms to the right. Simplifying n = 0Subproblem 2
Set the factor '(-3 + -7n + 5n2)' equal to zero and attempt to solve: Simplifying -3 + -7n + 5n2 = 0 Solving -3 + -7n + 5n2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.6 + -1.4n + n2 = 0 Move the constant term to the right: Add '0.6' to each side of the equation. -0.6 + -1.4n + 0.6 + n2 = 0 + 0.6 Reorder the terms: -0.6 + 0.6 + -1.4n + n2 = 0 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + -1.4n + n2 = 0 + 0.6 -1.4n + n2 = 0 + 0.6 Combine like terms: 0 + 0.6 = 0.6 -1.4n + n2 = 0.6 The n term is -1.4n. Take half its coefficient (-0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. -1.4n + 0.49 + n2 = 0.6 + 0.49 Reorder the terms: 0.49 + -1.4n + n2 = 0.6 + 0.49 Combine like terms: 0.6 + 0.49 = 1.09 0.49 + -1.4n + n2 = 1.09 Factor a perfect square on the left side: (n + -0.7)(n + -0.7) = 1.09 Calculate the square root of the right side: 1.044030651 Break this problem into two subproblems by setting (n + -0.7) equal to 1.044030651 and -1.044030651.Subproblem 1
n + -0.7 = 1.044030651 Simplifying n + -0.7 = 1.044030651 Reorder the terms: -0.7 + n = 1.044030651 Solving -0.7 + n = 1.044030651 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + n = 1.044030651 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + n = 1.044030651 + 0.7 n = 1.044030651 + 0.7 Combine like terms: 1.044030651 + 0.7 = 1.744030651 n = 1.744030651 Simplifying n = 1.744030651Subproblem 2
n + -0.7 = -1.044030651 Simplifying n + -0.7 = -1.044030651 Reorder the terms: -0.7 + n = -1.044030651 Solving -0.7 + n = -1.044030651 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.7' to each side of the equation. -0.7 + 0.7 + n = -1.044030651 + 0.7 Combine like terms: -0.7 + 0.7 = 0.0 0.0 + n = -1.044030651 + 0.7 n = -1.044030651 + 0.7 Combine like terms: -1.044030651 + 0.7 = -0.344030651 n = -0.344030651 Simplifying n = -0.344030651Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.744030651, -0.344030651}Solution
n = {0, 1.744030651, -0.344030651}
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