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Simplifying 6x2 + 6x + 1 = 22 Reorder the terms: 1 + 6x + 6x2 = 22 Solving 1 + 6x + 6x2 = 22 Solving for variable 'x'. Reorder the terms: 1 + -22 + 6x + 6x2 = 22 + -22 Combine like terms: 1 + -22 = -21 -21 + 6x + 6x2 = 22 + -22 Combine like terms: 22 + -22 = 0 -21 + 6x + 6x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-7 + 2x + 2x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-7 + 2x + 2x2)' equal to zero and attempt to solve: Simplifying -7 + 2x + 2x2 = 0 Solving -7 + 2x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + x + x2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + x + 3.5 + x2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + x + x2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x + x2 = 0 + 3.5 x + x2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 x + x2 = 3.5 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 = 3.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 3.5 + 0.25 Combine like terms: 3.5 + 0.25 = 3.75 1.25 + x2 = 3.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (x + 0.5) equal to 1.936491673 and -1.936491673.Subproblem 1
x + 0.5 = 1.936491673 Simplifying x + 0.5 = 1.936491673 Reorder the terms: 0.5 + x = 1.936491673 Solving 0.5 + x = 1.936491673 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 = 1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.936491673 + -0.5 x = 1.936491673 + -0.5 Combine like terms: 1.936491673 + -0.5 = 1.436491673 x = 1.436491673 Simplifying x = 1.436491673Subproblem 2
x + 0.5 = -1.936491673 Simplifying x + 0.5 = -1.936491673 Reorder the terms: 0.5 + x = -1.936491673 Solving 0.5 + x = -1.936491673 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 = -1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.936491673 + -0.5 x = -1.936491673 + -0.5 Combine like terms: -1.936491673 + -0.5 = -2.436491673 x = -2.436491673 Simplifying x = -2.436491673Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.436491673, -2.436491673}Solution
x = {1.436491673, -2.436491673}
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