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Simplifying 7 = -2x2 + 10x Reorder the terms: 7 = 10x + -2x2 Solving 7 = 10x + -2x2 Solving for variable 'x'. Reorder the terms: 7 + -10x + 2x2 = 10x + -10x + -2x2 + 2x2 Combine like terms: 10x + -10x = 0 7 + -10x + 2x2 = 0 + -2x2 + 2x2 7 + -10x + 2x2 = -2x2 + 2x2 Combine like terms: -2x2 + 2x2 = 0 7 + -10x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 3.5 + -5x + x2 = 0 Move the constant term to the right: Add '-3.5' to each side of the equation. 3.5 + -5x + -3.5 + x2 = 0 + -3.5 Reorder the terms: 3.5 + -3.5 + -5x + x2 = 0 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + -5x + x2 = 0 + -3.5 -5x + x2 = 0 + -3.5 Combine like terms: 0 + -3.5 = -3.5 -5x + x2 = -3.5 The x term is -5x. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5x + 6.25 + x2 = -3.5 + 6.25 Reorder the terms: 6.25 + -5x + x2 = -3.5 + 6.25 Combine like terms: -3.5 + 6.25 = 2.75 6.25 + -5x + x2 = 2.75 Factor a perfect square on the left side: (x + -2.5)(x + -2.5) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (x + -2.5) equal to 1.658312395 and -1.658312395.Subproblem 1
x + -2.5 = 1.658312395 Simplifying x + -2.5 = 1.658312395 Reorder the terms: -2.5 + x = 1.658312395 Solving -2.5 + x = 1.658312395 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 1.658312395 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 1.658312395 + 2.5 x = 1.658312395 + 2.5 Combine like terms: 1.658312395 + 2.5 = 4.158312395 x = 4.158312395 Simplifying x = 4.158312395Subproblem 2
x + -2.5 = -1.658312395 Simplifying x + -2.5 = -1.658312395 Reorder the terms: -2.5 + x = -1.658312395 Solving -2.5 + x = -1.658312395 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -1.658312395 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -1.658312395 + 2.5 x = -1.658312395 + 2.5 Combine like terms: -1.658312395 + 2.5 = 0.841687605 x = 0.841687605 Simplifying x = 0.841687605Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.158312395, 0.841687605}
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