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Simplifying 40x3 = (2x)(x + -4)(x + -1) Remove parenthesis around (2x) 40x3 = 2x(x + -4)(x + -1) Reorder the terms: 40x3 = 2x(-4 + x)(x + -1) Reorder the terms: 40x3 = 2x(-4 + x)(-1 + x) Multiply (-4 + x) * (-1 + x) 40x3 = 2x(-4(-1 + x) + x(-1 + x)) 40x3 = 2x((-1 * -4 + x * -4) + x(-1 + x)) 40x3 = 2x((4 + -4x) + x(-1 + x)) 40x3 = 2x(4 + -4x + (-1 * x + x * x)) 40x3 = 2x(4 + -4x + (-1x + x2)) Combine like terms: -4x + -1x = -5x 40x3 = 2x(4 + -5x + x2) 40x3 = (4 * 2x + -5x * 2x + x2 * 2x) 40x3 = (8x + -10x2 + 2x3) Solving 40x3 = 8x + -10x2 + 2x3 Solving for variable 'x'. Reorder the terms: -8x + 10x2 + 40x3 + -2x3 = 8x + -10x2 + 2x3 + -8x + 10x2 + -2x3 Combine like terms: 40x3 + -2x3 = 38x3 -8x + 10x2 + 38x3 = 8x + -10x2 + 2x3 + -8x + 10x2 + -2x3 Reorder the terms: -8x + 10x2 + 38x3 = 8x + -8x + -10x2 + 10x2 + 2x3 + -2x3 Combine like terms: 8x + -8x = 0 -8x + 10x2 + 38x3 = 0 + -10x2 + 10x2 + 2x3 + -2x3 -8x + 10x2 + 38x3 = -10x2 + 10x2 + 2x3 + -2x3 Combine like terms: -10x2 + 10x2 = 0 -8x + 10x2 + 38x3 = 0 + 2x3 + -2x3 -8x + 10x2 + 38x3 = 2x3 + -2x3 Combine like terms: 2x3 + -2x3 = 0 -8x + 10x2 + 38x3 = 0 Factor out the Greatest Common Factor (GCF), '2x'. 2x(-4 + 5x + 19x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(-4 + 5x + 19x2)' equal to zero and attempt to solve: Simplifying -4 + 5x + 19x2 = 0 Solving -4 + 5x + 19x2 = 0 Begin completing the square. Divide all terms by 19 the coefficient of the squared term: Divide each side by '19'. -0.2105263158 + 0.2631578947x + x2 = 0 Move the constant term to the right: Add '0.2105263158' to each side of the equation. -0.2105263158 + 0.2631578947x + 0.2105263158 + x2 = 0 + 0.2105263158 Reorder the terms: -0.2105263158 + 0.2105263158 + 0.2631578947x + x2 = 0 + 0.2105263158 Combine like terms: -0.2105263158 + 0.2105263158 = 0.0000000000 0.0000000000 + 0.2631578947x + x2 = 0 + 0.2105263158 0.2631578947x + x2 = 0 + 0.2105263158 Combine like terms: 0 + 0.2105263158 = 0.2105263158 0.2631578947x + x2 = 0.2105263158 The x term is 0.2631578947x. Take half its coefficient (0.1315789474). Square it (0.01731301940) and add it to both sides. Add '0.01731301940' to each side of the equation. 0.2631578947x + 0.01731301940 + x2 = 0.2105263158 + 0.01731301940 Reorder the terms: 0.01731301940 + 0.2631578947x + x2 = 0.2105263158 + 0.01731301940 Combine like terms: 0.2105263158 + 0.01731301940 = 0.2278393352 0.01731301940 + 0.2631578947x + x2 = 0.2278393352 Factor a perfect square on the left side: (x + 0.1315789474)(x + 0.1315789474) = 0.2278393352 Calculate the square root of the right side: 0.477325188 Break this problem into two subproblems by setting (x + 0.1315789474) equal to 0.477325188 and -0.477325188.Subproblem 1
x + 0.1315789474 = 0.477325188 Simplifying x + 0.1315789474 = 0.477325188 Reorder the terms: 0.1315789474 + x = 0.477325188 Solving 0.1315789474 + x = 0.477325188 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1315789474' to each side of the equation. 0.1315789474 + -0.1315789474 + x = 0.477325188 + -0.1315789474 Combine like terms: 0.1315789474 + -0.1315789474 = 0.0000000000 0.0000000000 + x = 0.477325188 + -0.1315789474 x = 0.477325188 + -0.1315789474 Combine like terms: 0.477325188 + -0.1315789474 = 0.3457462406 x = 0.3457462406 Simplifying x = 0.3457462406Subproblem 2
x + 0.1315789474 = -0.477325188 Simplifying x + 0.1315789474 = -0.477325188 Reorder the terms: 0.1315789474 + x = -0.477325188 Solving 0.1315789474 + x = -0.477325188 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1315789474' to each side of the equation. 0.1315789474 + -0.1315789474 + x = -0.477325188 + -0.1315789474 Combine like terms: 0.1315789474 + -0.1315789474 = 0.0000000000 0.0000000000 + x = -0.477325188 + -0.1315789474 x = -0.477325188 + -0.1315789474 Combine like terms: -0.477325188 + -0.1315789474 = -0.6089041354 x = -0.6089041354 Simplifying x = -0.6089041354Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.3457462406, -0.6089041354}Solution
x = {0, 0.3457462406, -0.6089041354}
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