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Simplifying 500 = 10x2 + 2x Reorder the terms: 500 = 2x + 10x2 Solving 500 = 2x + 10x2 Solving for variable 'x'. Reorder the terms: 500 + -2x + -10x2 = 2x + -2x + 10x2 + -10x2 Combine like terms: 2x + -2x = 0 500 + -2x + -10x2 = 0 + 10x2 + -10x2 500 + -2x + -10x2 = 10x2 + -10x2 Combine like terms: 10x2 + -10x2 = 0 500 + -2x + -10x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(250 + -1x + -5x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(250 + -1x + -5x2)' equal to zero and attempt to solve: Simplifying 250 + -1x + -5x2 = 0 Solving 250 + -1x + -5x2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -50 + 0.2x + x2 = 0 Move the constant term to the right: Add '50' to each side of the equation. -50 + 0.2x + 50 + x2 = 0 + 50 Reorder the terms: -50 + 50 + 0.2x + x2 = 0 + 50 Combine like terms: -50 + 50 = 0 0 + 0.2x + x2 = 0 + 50 0.2x + x2 = 0 + 50 Combine like terms: 0 + 50 = 50 0.2x + x2 = 50 The x term is 0.2x. Take half its coefficient (0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. 0.2x + 0.01 + x2 = 50 + 0.01 Reorder the terms: 0.01 + 0.2x + x2 = 50 + 0.01 Combine like terms: 50 + 0.01 = 50.01 0.01 + 0.2x + x2 = 50.01 Factor a perfect square on the left side: (x + 0.1)(x + 0.1) = 50.01 Calculate the square root of the right side: 7.071774883 Break this problem into two subproblems by setting (x + 0.1) equal to 7.071774883 and -7.071774883.Subproblem 1
x + 0.1 = 7.071774883 Simplifying x + 0.1 = 7.071774883 Reorder the terms: 0.1 + x = 7.071774883 Solving 0.1 + x = 7.071774883 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = 7.071774883 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = 7.071774883 + -0.1 x = 7.071774883 + -0.1 Combine like terms: 7.071774883 + -0.1 = 6.971774883 x = 6.971774883 Simplifying x = 6.971774883Subproblem 2
x + 0.1 = -7.071774883 Simplifying x + 0.1 = -7.071774883 Reorder the terms: 0.1 + x = -7.071774883 Solving 0.1 + x = -7.071774883 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1' to each side of the equation. 0.1 + -0.1 + x = -7.071774883 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + x = -7.071774883 + -0.1 x = -7.071774883 + -0.1 Combine like terms: -7.071774883 + -0.1 = -7.171774883 x = -7.171774883 Simplifying x = -7.171774883Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.971774883, -7.171774883}Solution
x = {6.971774883, -7.171774883}
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