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Simplifying 5(4x2 + 2x + 2) + -2(6x2 + 4x + 7) = 0 Reorder the terms: 5(2 + 2x + 4x2) + -2(6x2 + 4x + 7) = 0 (2 * 5 + 2x * 5 + 4x2 * 5) + -2(6x2 + 4x + 7) = 0 (10 + 10x + 20x2) + -2(6x2 + 4x + 7) = 0 Reorder the terms: 10 + 10x + 20x2 + -2(7 + 4x + 6x2) = 0 10 + 10x + 20x2 + (7 * -2 + 4x * -2 + 6x2 * -2) = 0 10 + 10x + 20x2 + (-14 + -8x + -12x2) = 0 Reorder the terms: 10 + -14 + 10x + -8x + 20x2 + -12x2 = 0 Combine like terms: 10 + -14 = -4 -4 + 10x + -8x + 20x2 + -12x2 = 0 Combine like terms: 10x + -8x = 2x -4 + 2x + 20x2 + -12x2 = 0 Combine like terms: 20x2 + -12x2 = 8x2 -4 + 2x + 8x2 = 0 Solving -4 + 2x + 8x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-2 + x + 4x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-2 + x + 4x2)' equal to zero and attempt to solve: Simplifying -2 + x + 4x2 = 0 Solving -2 + x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.5 + 0.25x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 0.25x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 0.25x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 0.25x + x2 = 0 + 0.5 0.25x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 0.25x + x2 = 0.5 The x term is x. 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. 0.25x + 0.25 + x2 = 0.5 + 0.25 Reorder the terms: 0.25 + 0.25x + x2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 0.25 + 0.25x + x2 = 0.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (x + 0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
x + 0.5 = 0.866025404 Simplifying x + 0.5 = 0.866025404 Reorder the terms: 0.5 + x = 0.866025404 Solving 0.5 + x = 0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.866025404 + -0.5 x = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 x = 0.366025404 Simplifying x = 0.366025404Subproblem 2
x + 0.5 = -0.866025404 Simplifying x + 0.5 = -0.866025404 Reorder the terms: 0.5 + x = -0.866025404 Solving 0.5 + x = -0.866025404 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.866025404 + -0.5 x = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 x = -1.366025404 Simplifying x = -1.366025404Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.366025404, -1.366025404}Solution
x = {0.366025404, -1.366025404}
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