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Simplifying 7 = (X * 0.25) + [(X * 0.25) * 0.065] * X Reorder the terms for easier multiplication: 7 = (0.25X) + [(X * 0.25) * 0.065] * X Reorder the terms for easier multiplication: 7 = (0.25X) + [(0.25X) * 0.065] * X Remove parenthesis around (0.25X) 7 = (0.25X) + [0.25X * 0.065] * X Reorder the terms for easier multiplication: 7 = (0.25X) + [0.25 * 0.065X] * X Multiply 0.25 * 0.065 7 = (0.25X) + [0.01625X] * X Remove brackets around [0.01625X] 7 = (0.25X) + 0.01625X * X Multiply X * X 7 = (0.25X) + 0.01625X2 Solving 7 = (0.25X) + 0.01625X2 Solving for variable 'X'. Reorder the terms: 7 + (-0.25X) + -0.01625X2 = (0.25X) + (-0.25X) + 0.01625X2 + -0.01625X2 Combine like terms: (0.25X) + (-0.25X) = 0.00 7 + (-0.25X) + -0.01625X2 = 0.00 + 0.01625X2 + -0.01625X2 7 + (-0.25X) + -0.01625X2 = 0.01625X2 + -0.01625X2 Combine like terms: 0.01625X2 + -0.01625X2 = 0.00000 7 + (-0.25X) + -0.01625X2 = 0.00000 Begin completing the square. Divide all terms by -0.01625 the coefficient of the squared term: Divide each side by '-0.01625'. -430.7692308 + (15.38461538X) + X2 = 0 Move the constant term to the right: Add '430.7692308' to each side of the equation. -430.7692308 + (15.38461538X) + 430.7692308 + X2 = 0 + 430.7692308 Reorder the terms: -430.7692308 + 430.7692308 + (15.38461538X) + X2 = 0 + 430.7692308 Combine like terms: -430.7692308 + 430.7692308 = 0.0000000 0.0000000 + (15.38461538X) + X2 = 0 + 430.7692308 (15.38461538X) + X2 = 0 + 430.7692308 Combine like terms: 0 + 430.7692308 = 430.7692308 (15.38461538X) + X2 = 430.7692308 The X term is (15.38461538X). Take half its coefficient (7.69230769). Square it (59.17159760) and add it to both sides. Add '59.17159760' to each side of the equation. (15.38461538X) + 59.17159760 + X2 = 430.7692308 + 59.17159760 Reorder the terms: 59.17159760 + (15.38461538X) + X2 = 430.7692308 + 59.17159760 Combine like terms: 430.7692308 + 59.17159760 = 489.9408284 59.17159760 + (15.38461538X) + X2 = 489.9408284 Factor a perfect square on the left side: (X + 7.69230769)(X + 7.69230769) = 489.9408284 Calculate the square root of the right side: 22.134607031 Break this problem into two subproblems by setting (X + 7.69230769) equal to 22.134607031 and -22.134607031.Subproblem 1
X + 7.69230769 = 22.134607031 Simplifying X + 7.69230769 = 22.134607031 Reorder the terms: 7.69230769 + X = 22.134607031 Solving 7.69230769 + X = 22.134607031 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-7.69230769' to each side of the equation. 7.69230769 + -7.69230769 + X = 22.134607031 + -7.69230769 Combine like terms: 7.69230769 + -7.69230769 = 0.00000000 0.00000000 + X = 22.134607031 + -7.69230769 X = 22.134607031 + -7.69230769 Combine like terms: 22.134607031 + -7.69230769 = 14.442299341 X = 14.442299341 Simplifying X = 14.442299341Subproblem 2
X + 7.69230769 = -22.134607031 Simplifying X + 7.69230769 = -22.134607031 Reorder the terms: 7.69230769 + X = -22.134607031 Solving 7.69230769 + X = -22.134607031 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-7.69230769' to each side of the equation. 7.69230769 + -7.69230769 + X = -22.134607031 + -7.69230769 Combine like terms: 7.69230769 + -7.69230769 = 0.00000000 0.00000000 + X = -22.134607031 + -7.69230769 X = -22.134607031 + -7.69230769 Combine like terms: -22.134607031 + -7.69230769 = -29.826914721 X = -29.826914721 Simplifying X = -29.826914721Solution
The solution to the problem is based on the solutions from the subproblems. X = {14.442299341, -29.826914721}
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