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Simplifying 4x2 + 6x + 2 = 91 Reorder the terms: 2 + 6x + 4x2 = 91 Solving 2 + 6x + 4x2 = 91 Solving for variable 'x'. Reorder the terms: 2 + -91 + 6x + 4x2 = 91 + -91 Combine like terms: 2 + -91 = -89 -89 + 6x + 4x2 = 91 + -91 Combine like terms: 91 + -91 = 0 -89 + 6x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -22.25 + 1.5x + x2 = 0 Move the constant term to the right: Add '22.25' to each side of the equation. -22.25 + 1.5x + 22.25 + x2 = 0 + 22.25 Reorder the terms: -22.25 + 22.25 + 1.5x + x2 = 0 + 22.25 Combine like terms: -22.25 + 22.25 = 0.00 0.00 + 1.5x + x2 = 0 + 22.25 1.5x + x2 = 0 + 22.25 Combine like terms: 0 + 22.25 = 22.25 1.5x + x2 = 22.25 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 22.25 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 22.25 + 0.5625 Combine like terms: 22.25 + 0.5625 = 22.8125 0.5625 + 1.5x + x2 = 22.8125 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 22.8125 Calculate the square root of the right side: 4.776243294 Break this problem into two subproblems by setting (x + 0.75) equal to 4.776243294 and -4.776243294.Subproblem 1
x + 0.75 = 4.776243294 Simplifying x + 0.75 = 4.776243294 Reorder the terms: 0.75 + x = 4.776243294 Solving 0.75 + x = 4.776243294 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 4.776243294 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 4.776243294 + -0.75 x = 4.776243294 + -0.75 Combine like terms: 4.776243294 + -0.75 = 4.026243294 x = 4.026243294 Simplifying x = 4.026243294Subproblem 2
x + 0.75 = -4.776243294 Simplifying x + 0.75 = -4.776243294 Reorder the terms: 0.75 + x = -4.776243294 Solving 0.75 + x = -4.776243294 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -4.776243294 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -4.776243294 + -0.75 x = -4.776243294 + -0.75 Combine like terms: -4.776243294 + -0.75 = -5.526243294 x = -5.526243294 Simplifying x = -5.526243294Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.026243294, -5.526243294}
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