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Simplifying 40 = -16x2 + 112x + 4 Reorder the terms: 40 = 4 + 112x + -16x2 Solving 40 = 4 + 112x + -16x2 Solving for variable 'x'. Combine like terms: 40 + -4 = 36 36 + -112x + 16x2 = 4 + 112x + -16x2 + -4 + -112x + 16x2 Reorder the terms: 36 + -112x + 16x2 = 4 + -4 + 112x + -112x + -16x2 + 16x2 Combine like terms: 4 + -4 = 0 36 + -112x + 16x2 = 0 + 112x + -112x + -16x2 + 16x2 36 + -112x + 16x2 = 112x + -112x + -16x2 + 16x2 Combine like terms: 112x + -112x = 0 36 + -112x + 16x2 = 0 + -16x2 + 16x2 36 + -112x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 36 + -112x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(9 + -28x + 4x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(9 + -28x + 4x2)' equal to zero and attempt to solve: Simplifying 9 + -28x + 4x2 = 0 Solving 9 + -28x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 2.25 + -7x + x2 = 0 Move the constant term to the right: Add '-2.25' to each side of the equation. 2.25 + -7x + -2.25 + x2 = 0 + -2.25 Reorder the terms: 2.25 + -2.25 + -7x + x2 = 0 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + -7x + x2 = 0 + -2.25 -7x + x2 = 0 + -2.25 Combine like terms: 0 + -2.25 = -2.25 -7x + x2 = -2.25 The x term is -7x. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7x + 12.25 + x2 = -2.25 + 12.25 Reorder the terms: 12.25 + -7x + x2 = -2.25 + 12.25 Combine like terms: -2.25 + 12.25 = 10 12.25 + -7x + x2 = 10 Factor a perfect square on the left side: (x + -3.5)(x + -3.5) = 10 Calculate the square root of the right side: 3.16227766 Break this problem into two subproblems by setting (x + -3.5) equal to 3.16227766 and -3.16227766.Subproblem 1
x + -3.5 = 3.16227766 Simplifying x + -3.5 = 3.16227766 Reorder the terms: -3.5 + x = 3.16227766 Solving -3.5 + x = 3.16227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 3.16227766 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 3.16227766 + 3.5 x = 3.16227766 + 3.5 Combine like terms: 3.16227766 + 3.5 = 6.66227766 x = 6.66227766 Simplifying x = 6.66227766Subproblem 2
x + -3.5 = -3.16227766 Simplifying x + -3.5 = -3.16227766 Reorder the terms: -3.5 + x = -3.16227766 Solving -3.5 + x = -3.16227766 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -3.16227766 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -3.16227766 + 3.5 x = -3.16227766 + 3.5 Combine like terms: -3.16227766 + 3.5 = 0.33772234 x = 0.33772234 Simplifying x = 0.33772234Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.66227766, 0.33772234}Solution
x = {6.66227766, 0.33772234}
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