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Simplifying 0 = 45x2 + -72x + 24 Reorder the terms: 0 = 24 + -72x + 45x2 Solving 0 = 24 + -72x + 45x2 Solving for variable 'x'. Combine like terms: 0 + -24 = -24 -24 + 72x + -45x2 = 24 + -72x + 45x2 + -24 + 72x + -45x2 Reorder the terms: -24 + 72x + -45x2 = 24 + -24 + -72x + 72x + 45x2 + -45x2 Combine like terms: 24 + -24 = 0 -24 + 72x + -45x2 = 0 + -72x + 72x + 45x2 + -45x2 -24 + 72x + -45x2 = -72x + 72x + 45x2 + -45x2 Combine like terms: -72x + 72x = 0 -24 + 72x + -45x2 = 0 + 45x2 + -45x2 -24 + 72x + -45x2 = 45x2 + -45x2 Combine like terms: 45x2 + -45x2 = 0 -24 + 72x + -45x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-8 + 24x + -15x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-8 + 24x + -15x2)' equal to zero and attempt to solve: Simplifying -8 + 24x + -15x2 = 0 Solving -8 + 24x + -15x2 = 0 Begin completing the square. Divide all terms by -15 the coefficient of the squared term: Divide each side by '-15'. 0.5333333333 + -1.6x + x2 = 0 Move the constant term to the right: Add '-0.5333333333' to each side of the equation. 0.5333333333 + -1.6x + -0.5333333333 + x2 = 0 + -0.5333333333 Reorder the terms: 0.5333333333 + -0.5333333333 + -1.6x + x2 = 0 + -0.5333333333 Combine like terms: 0.5333333333 + -0.5333333333 = 0.0000000000 0.0000000000 + -1.6x + x2 = 0 + -0.5333333333 -1.6x + x2 = 0 + -0.5333333333 Combine like terms: 0 + -0.5333333333 = -0.5333333333 -1.6x + x2 = -0.5333333333 The x term is -1.6x. Take half its coefficient (-0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. -1.6x + 0.64 + x2 = -0.5333333333 + 0.64 Reorder the terms: 0.64 + -1.6x + x2 = -0.5333333333 + 0.64 Combine like terms: -0.5333333333 + 0.64 = 0.1066666667 0.64 + -1.6x + x2 = 0.1066666667 Factor a perfect square on the left side: (x + -0.8)(x + -0.8) = 0.1066666667 Calculate the square root of the right side: 0.326598632 Break this problem into two subproblems by setting (x + -0.8) equal to 0.326598632 and -0.326598632.Subproblem 1
x + -0.8 = 0.326598632 Simplifying x + -0.8 = 0.326598632 Reorder the terms: -0.8 + x = 0.326598632 Solving -0.8 + x = 0.326598632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8' to each side of the equation. -0.8 + 0.8 + x = 0.326598632 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + x = 0.326598632 + 0.8 x = 0.326598632 + 0.8 Combine like terms: 0.326598632 + 0.8 = 1.126598632 x = 1.126598632 Simplifying x = 1.126598632Subproblem 2
x + -0.8 = -0.326598632 Simplifying x + -0.8 = -0.326598632 Reorder the terms: -0.8 + x = -0.326598632 Solving -0.8 + x = -0.326598632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8' to each side of the equation. -0.8 + 0.8 + x = -0.326598632 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + x = -0.326598632 + 0.8 x = -0.326598632 + 0.8 Combine like terms: -0.326598632 + 0.8 = 0.473401368 x = 0.473401368 Simplifying x = 0.473401368Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.126598632, 0.473401368}Solution
x = {1.126598632, 0.473401368}
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