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Simplifying 4x + 24x + x2 + -5x2 = 1 Combine like terms: 4x + 24x = 28x 28x + x2 + -5x2 = 1 Combine like terms: x2 + -5x2 = -4x2 28x + -4x2 = 1 Solving 28x + -4x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + 28x + -4x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 28x + -4x2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 0.25 + -7x + x2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -7x + -0.25 + x2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -7x + x2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -7x + x2 = 0 + -0.25 -7x + x2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -7x + x2 = -0.25 The x term is -7x. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7x + 12.25 + x2 = -0.25 + 12.25 Reorder the terms: 12.25 + -7x + x2 = -0.25 + 12.25 Combine like terms: -0.25 + 12.25 = 12 12.25 + -7x + x2 = 12 Factor a perfect square on the left side: (x + -3.5)(x + -3.5) = 12 Calculate the square root of the right side: 3.464101615 Break this problem into two subproblems by setting (x + -3.5) equal to 3.464101615 and -3.464101615.Subproblem 1
x + -3.5 = 3.464101615 Simplifying x + -3.5 = 3.464101615 Reorder the terms: -3.5 + x = 3.464101615 Solving -3.5 + x = 3.464101615 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 3.464101615 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 3.464101615 + 3.5 x = 3.464101615 + 3.5 Combine like terms: 3.464101615 + 3.5 = 6.964101615 x = 6.964101615 Simplifying x = 6.964101615Subproblem 2
x + -3.5 = -3.464101615 Simplifying x + -3.5 = -3.464101615 Reorder the terms: -3.5 + x = -3.464101615 Solving -3.5 + x = -3.464101615 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -3.464101615 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -3.464101615 + 3.5 x = -3.464101615 + 3.5 Combine like terms: -3.464101615 + 3.5 = 0.035898385 x = 0.035898385 Simplifying x = 0.035898385Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.964101615, 0.035898385}
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