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Simplifying 4x2 + 1 = 7x Reorder the terms: 1 + 4x2 = 7x Solving 1 + 4x2 = 7x Solving for variable 'x'. Reorder the terms: 1 + -7x + 4x2 = 7x + -7x Combine like terms: 7x + -7x = 0 1 + -7x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + -1.75x + x2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -1.75x + -0.25 + x2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -1.75x + x2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -1.75x + x2 = 0 + -0.25 -1.75x + x2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -1.75x + x2 = -0.25 The x term is -1.75x. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75x + 0.765625 + x2 = -0.25 + 0.765625 Reorder the terms: 0.765625 + -1.75x + x2 = -0.25 + 0.765625 Combine like terms: -0.25 + 0.765625 = 0.515625 0.765625 + -1.75x + x2 = 0.515625 Factor a perfect square on the left side: (x + -0.875)(x + -0.875) = 0.515625 Calculate the square root of the right side: 0.718070331 Break this problem into two subproblems by setting (x + -0.875) equal to 0.718070331 and -0.718070331.Subproblem 1
x + -0.875 = 0.718070331 Simplifying x + -0.875 = 0.718070331 Reorder the terms: -0.875 + x = 0.718070331 Solving -0.875 + x = 0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = 0.718070331 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = 0.718070331 + 0.875 x = 0.718070331 + 0.875 Combine like terms: 0.718070331 + 0.875 = 1.593070331 x = 1.593070331 Simplifying x = 1.593070331Subproblem 2
x + -0.875 = -0.718070331 Simplifying x + -0.875 = -0.718070331 Reorder the terms: -0.875 + x = -0.718070331 Solving -0.875 + x = -0.718070331 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = -0.718070331 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = -0.718070331 + 0.875 x = -0.718070331 + 0.875 Combine like terms: -0.718070331 + 0.875 = 0.156929669 x = 0.156929669 Simplifying x = 0.156929669Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.593070331, 0.156929669}
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