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Simplifying 0 = 24x2 + 28x + -24 Reorder the terms: 0 = -24 + 28x + 24x2 Solving 0 = -24 + 28x + 24x2 Solving for variable 'x'. Combine like terms: 0 + 24 = 24 24 + -28x + -24x2 = -24 + 28x + 24x2 + 24 + -28x + -24x2 Reorder the terms: 24 + -28x + -24x2 = -24 + 24 + 28x + -28x + 24x2 + -24x2 Combine like terms: -24 + 24 = 0 24 + -28x + -24x2 = 0 + 28x + -28x + 24x2 + -24x2 24 + -28x + -24x2 = 28x + -28x + 24x2 + -24x2 Combine like terms: 28x + -28x = 0 24 + -28x + -24x2 = 0 + 24x2 + -24x2 24 + -28x + -24x2 = 24x2 + -24x2 Combine like terms: 24x2 + -24x2 = 0 24 + -28x + -24x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(6 + -7x + -6x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(6 + -7x + -6x2)' equal to zero and attempt to solve: Simplifying 6 + -7x + -6x2 = 0 Solving 6 + -7x + -6x2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1 + 1.166666667x + x2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 1.166666667x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + 1.166666667x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 1.166666667x + x2 = 0 + 1 1.166666667x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 1.166666667x + x2 = 1 The x term is 1.166666667x. Take half its coefficient (0.5833333335). Square it (0.3402777780) and add it to both sides. Add '0.3402777780' to each side of the equation. 1.166666667x + 0.3402777780 + x2 = 1 + 0.3402777780 Reorder the terms: 0.3402777780 + 1.166666667x + x2 = 1 + 0.3402777780 Combine like terms: 1 + 0.3402777780 = 1.340277778 0.3402777780 + 1.166666667x + x2 = 1.340277778 Factor a perfect square on the left side: (x + 0.5833333335)(x + 0.5833333335) = 1.340277778 Calculate the square root of the right side: 1.157703666 Break this problem into two subproblems by setting (x + 0.5833333335) equal to 1.157703666 and -1.157703666.Subproblem 1
x + 0.5833333335 = 1.157703666 Simplifying x + 0.5833333335 = 1.157703666 Reorder the terms: 0.5833333335 + x = 1.157703666 Solving 0.5833333335 + x = 1.157703666 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + x = 1.157703666 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + x = 1.157703666 + -0.5833333335 x = 1.157703666 + -0.5833333335 Combine like terms: 1.157703666 + -0.5833333335 = 0.5743703325 x = 0.5743703325 Simplifying x = 0.5743703325Subproblem 2
x + 0.5833333335 = -1.157703666 Simplifying x + 0.5833333335 = -1.157703666 Reorder the terms: 0.5833333335 + x = -1.157703666 Solving 0.5833333335 + x = -1.157703666 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + x = -1.157703666 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + x = -1.157703666 + -0.5833333335 x = -1.157703666 + -0.5833333335 Combine like terms: -1.157703666 + -0.5833333335 = -1.7410369995 x = -1.7410369995 Simplifying x = -1.7410369995Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.5743703325, -1.7410369995}Solution
x = {0.5743703325, -1.7410369995}
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