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Simplifying 78 = 16x2 + 32x Reorder the terms: 78 = 32x + 16x2 Solving 78 = 32x + 16x2 Solving for variable 'x'. Reorder the terms: 78 + -32x + -16x2 = 32x + -32x + 16x2 + -16x2 Combine like terms: 32x + -32x = 0 78 + -32x + -16x2 = 0 + 16x2 + -16x2 78 + -32x + -16x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 78 + -32x + -16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(39 + -16x + -8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(39 + -16x + -8x2)' equal to zero and attempt to solve: Simplifying 39 + -16x + -8x2 = 0 Solving 39 + -16x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -4.875 + 2x + x2 = 0 Move the constant term to the right: Add '4.875' to each side of the equation. -4.875 + 2x + 4.875 + x2 = 0 + 4.875 Reorder the terms: -4.875 + 4.875 + 2x + x2 = 0 + 4.875 Combine like terms: -4.875 + 4.875 = 0.000 0.000 + 2x + x2 = 0 + 4.875 2x + x2 = 0 + 4.875 Combine like terms: 0 + 4.875 = 4.875 2x + x2 = 4.875 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 4.875 + 1 Reorder the terms: 1 + 2x + x2 = 4.875 + 1 Combine like terms: 4.875 + 1 = 5.875 1 + 2x + x2 = 5.875 Factor a perfect square on the left side: (x + 1)(x + 1) = 5.875 Calculate the square root of the right side: 2.423839929 Break this problem into two subproblems by setting (x + 1) equal to 2.423839929 and -2.423839929.Subproblem 1
x + 1 = 2.423839929 Simplifying x + 1 = 2.423839929 Reorder the terms: 1 + x = 2.423839929 Solving 1 + x = 2.423839929 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 2.423839929 + -1 Combine like terms: 1 + -1 = 0 0 + x = 2.423839929 + -1 x = 2.423839929 + -1 Combine like terms: 2.423839929 + -1 = 1.423839929 x = 1.423839929 Simplifying x = 1.423839929Subproblem 2
x + 1 = -2.423839929 Simplifying x + 1 = -2.423839929 Reorder the terms: 1 + x = -2.423839929 Solving 1 + x = -2.423839929 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -2.423839929 + -1 Combine like terms: 1 + -1 = 0 0 + x = -2.423839929 + -1 x = -2.423839929 + -1 Combine like terms: -2.423839929 + -1 = -3.423839929 x = -3.423839929 Simplifying x = -3.423839929Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.423839929, -3.423839929}Solution
x = {1.423839929, -3.423839929}
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