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Simplifying 9x2 + 10 + 3x + 62 = 180 Reorder the terms: 10 + 62 + 3x + 9x2 = 180 Combine like terms: 10 + 62 = 72 72 + 3x + 9x2 = 180 Solving 72 + 3x + 9x2 = 180 Solving for variable 'x'. Reorder the terms: 72 + -180 + 3x + 9x2 = 180 + -180 Combine like terms: 72 + -180 = -108 -108 + 3x + 9x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -108 + 3x + 9x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-36 + x + 3x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-36 + x + 3x2)' equal to zero and attempt to solve: Simplifying -36 + x + 3x2 = 0 Solving -36 + x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -12 + 0.3333333333x + x2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 0.3333333333x + 12 + x2 = 0 + 12 Reorder the terms: -12 + 12 + 0.3333333333x + x2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 0.3333333333x + x2 = 0 + 12 0.3333333333x + x2 = 0 + 12 Combine like terms: 0 + 12 = 12 0.3333333333x + x2 = 12 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. 0.3333333333x + 0.25 + x2 = 12 + 0.25 Reorder the terms: 0.25 + 0.3333333333x + x2 = 12 + 0.25 Combine like terms: 12 + 0.25 = 12.25 0.25 + 0.3333333333x + x2 = 12.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 12.25 Calculate the square root of the right side: 3.5 Break this problem into two subproblems by setting (x + 0.5) equal to 3.5 and -3.5.Subproblem 1
x + 0.5 = 3.5 Simplifying x + 0.5 = 3.5 Reorder the terms: 0.5 + x = 3.5 Solving 0.5 + x = 3.5 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 = 3.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 3.5 + -0.5 x = 3.5 + -0.5 Combine like terms: 3.5 + -0.5 = 3 x = 3 Simplifying x = 3Subproblem 2
x + 0.5 = -3.5 Simplifying x + 0.5 = -3.5 Reorder the terms: 0.5 + x = -3.5 Solving 0.5 + x = -3.5 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 = -3.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -3.5 + -0.5 x = -3.5 + -0.5 Combine like terms: -3.5 + -0.5 = -4 x = -4 Simplifying x = -4Solution
The solution to the problem is based on the solutions from the subproblems. x = {3, -4}Solution
x = {3, -4}
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