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Simplifying 5x(5x) + 6x + 5 = 32 Remove parenthesis around (5x) 5x * 5x + 6x + 5 = 32 Reorder the terms for easier multiplication: 5 * 5x * x + 6x + 5 = 32 Multiply 5 * 5 25x * x + 6x + 5 = 32 Multiply x * x 25x2 + 6x + 5 = 32 Reorder the terms: 5 + 6x + 25x2 = 32 Solving 5 + 6x + 25x2 = 32 Solving for variable 'x'. Reorder the terms: 5 + -32 + 6x + 25x2 = 32 + -32 Combine like terms: 5 + -32 = -27 -27 + 6x + 25x2 = 32 + -32 Combine like terms: 32 + -32 = 0 -27 + 6x + 25x2 = 0 Begin completing the square. Divide all terms by 25 the coefficient of the squared term: Divide each side by '25'. -1.08 + 0.24x + x2 = 0 Move the constant term to the right: Add '1.08' to each side of the equation. -1.08 + 0.24x + 1.08 + x2 = 0 + 1.08 Reorder the terms: -1.08 + 1.08 + 0.24x + x2 = 0 + 1.08 Combine like terms: -1.08 + 1.08 = 0.00 0.00 + 0.24x + x2 = 0 + 1.08 0.24x + x2 = 0 + 1.08 Combine like terms: 0 + 1.08 = 1.08 0.24x + x2 = 1.08 The x term is 0.24x. Take half its coefficient (0.12). Square it (0.0144) and add it to both sides. Add '0.0144' to each side of the equation. 0.24x + 0.0144 + x2 = 1.08 + 0.0144 Reorder the terms: 0.0144 + 0.24x + x2 = 1.08 + 0.0144 Combine like terms: 1.08 + 0.0144 = 1.0944 0.0144 + 0.24x + x2 = 1.0944 Factor a perfect square on the left side: (x + 0.12)(x + 0.12) = 1.0944 Calculate the square root of the right side: 1.046135746 Break this problem into two subproblems by setting (x + 0.12) equal to 1.046135746 and -1.046135746.Subproblem 1
x + 0.12 = 1.046135746 Simplifying x + 0.12 = 1.046135746 Reorder the terms: 0.12 + x = 1.046135746 Solving 0.12 + x = 1.046135746 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.12' to each side of the equation. 0.12 + -0.12 + x = 1.046135746 + -0.12 Combine like terms: 0.12 + -0.12 = 0.00 0.00 + x = 1.046135746 + -0.12 x = 1.046135746 + -0.12 Combine like terms: 1.046135746 + -0.12 = 0.926135746 x = 0.926135746 Simplifying x = 0.926135746Subproblem 2
x + 0.12 = -1.046135746 Simplifying x + 0.12 = -1.046135746 Reorder the terms: 0.12 + x = -1.046135746 Solving 0.12 + x = -1.046135746 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.12' to each side of the equation. 0.12 + -0.12 + x = -1.046135746 + -0.12 Combine like terms: 0.12 + -0.12 = 0.00 0.00 + x = -1.046135746 + -0.12 x = -1.046135746 + -0.12 Combine like terms: -1.046135746 + -0.12 = -1.166135746 x = -1.166135746 Simplifying x = -1.166135746Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.926135746, -1.166135746}
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