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Simplifying 5x2 + 1 = 10x Reorder the terms: 1 + 5x2 = 10x Solving 1 + 5x2 = 10x Solving for variable 'x'. Reorder the terms: 1 + -10x + 5x2 = 10x + -10x Combine like terms: 10x + -10x = 0 1 + -10x + 5x2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.2 + -2x + x2 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + -2x + -0.2 + x2 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + -2x + x2 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + -2x + x2 = 0 + -0.2 -2x + x2 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 -2x + x2 = -0.2 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = -0.2 + 1 Reorder the terms: 1 + -2x + x2 = -0.2 + 1 Combine like terms: -0.2 + 1 = 0.8 1 + -2x + x2 = 0.8 Factor a perfect square on the left side: (x + -1)(x + -1) = 0.8 Calculate the square root of the right side: 0.894427191 Break this problem into two subproblems by setting (x + -1) equal to 0.894427191 and -0.894427191.Subproblem 1
x + -1 = 0.894427191 Simplifying x + -1 = 0.894427191 Reorder the terms: -1 + x = 0.894427191 Solving -1 + x = 0.894427191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 0.894427191 + 1 Combine like terms: -1 + 1 = 0 0 + x = 0.894427191 + 1 x = 0.894427191 + 1 Combine like terms: 0.894427191 + 1 = 1.894427191 x = 1.894427191 Simplifying x = 1.894427191Subproblem 2
x + -1 = -0.894427191 Simplifying x + -1 = -0.894427191 Reorder the terms: -1 + x = -0.894427191 Solving -1 + x = -0.894427191 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -0.894427191 + 1 Combine like terms: -1 + 1 = 0 0 + x = -0.894427191 + 1 x = -0.894427191 + 1 Combine like terms: -0.894427191 + 1 = 0.105572809 x = 0.105572809 Simplifying x = 0.105572809Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.894427191, 0.105572809}
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