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Simplifying 6x2 + 6x + 5 = 32 Reorder the terms: 5 + 6x + 6x2 = 32 Solving 5 + 6x + 6x2 = 32 Solving for variable 'x'. Reorder the terms: 5 + -32 + 6x + 6x2 = 32 + -32 Combine like terms: 5 + -32 = -27 -27 + 6x + 6x2 = 32 + -32 Combine like terms: 32 + -32 = 0 -27 + 6x + 6x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-9 + 2x + 2x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-9 + 2x + 2x2)' equal to zero and attempt to solve: Simplifying -9 + 2x + 2x2 = 0 Solving -9 + 2x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -4.5 + x + x2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + x + 4.5 + x2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + x + x2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x + x2 = 0 + 4.5 x + x2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 x + x2 = 4.5 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + x2 = 4.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 4.5 + 0.25 Combine like terms: 4.5 + 0.25 = 4.75 1.25 + x2 = 4.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 4.75 Calculate the square root of the right side: 2.179449472 Break this problem into two subproblems by setting (x + 0.5) equal to 2.179449472 and -2.179449472.Subproblem 1
x + 0.5 = 2.179449472 Simplifying x + 0.5 = 2.179449472 Reorder the terms: 0.5 + x = 2.179449472 Solving 0.5 + x = 2.179449472 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 2.179449472 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 2.179449472 + -0.5 x = 2.179449472 + -0.5 Combine like terms: 2.179449472 + -0.5 = 1.679449472 x = 1.679449472 Simplifying x = 1.679449472Subproblem 2
x + 0.5 = -2.179449472 Simplifying x + 0.5 = -2.179449472 Reorder the terms: 0.5 + x = -2.179449472 Solving 0.5 + x = -2.179449472 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -2.179449472 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -2.179449472 + -0.5 x = -2.179449472 + -0.5 Combine like terms: -2.179449472 + -0.5 = -2.679449472 x = -2.679449472 Simplifying x = -2.679449472Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.679449472, -2.679449472}Solution
x = {1.679449472, -2.679449472}
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