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Simplifying 7d2 + 10(1 + -6d2) + -4d = 0 7d2 + (1 * 10 + -6d2 * 10) + -4d = 0 7d2 + (10 + -60d2) + -4d = 0 Reorder the terms: 10 + -4d + 7d2 + -60d2 = 0 Combine like terms: 7d2 + -60d2 = -53d2 10 + -4d + -53d2 = 0 Solving 10 + -4d + -53d2 = 0 Solving for variable 'd'. Begin completing the square. Divide all terms by -53 the coefficient of the squared term: Divide each side by '-53'. -0.1886792453 + 0.07547169811d + d2 = 0 Move the constant term to the right: Add '0.1886792453' to each side of the equation. -0.1886792453 + 0.07547169811d + 0.1886792453 + d2 = 0 + 0.1886792453 Reorder the terms: -0.1886792453 + 0.1886792453 + 0.07547169811d + d2 = 0 + 0.1886792453 Combine like terms: -0.1886792453 + 0.1886792453 = 0.0000000000 0.0000000000 + 0.07547169811d + d2 = 0 + 0.1886792453 0.07547169811d + d2 = 0 + 0.1886792453 Combine like terms: 0 + 0.1886792453 = 0.1886792453 0.07547169811d + d2 = 0.1886792453 The d term is 0.07547169811d. Take half its coefficient (0.03773584906). Square it (0.001423994304) and add it to both sides. Add '0.001423994304' to each side of the equation. 0.07547169811d + 0.001423994304 + d2 = 0.1886792453 + 0.001423994304 Reorder the terms: 0.001423994304 + 0.07547169811d + d2 = 0.1886792453 + 0.001423994304 Combine like terms: 0.1886792453 + 0.001423994304 = 0.190103239604 0.001423994304 + 0.07547169811d + d2 = 0.190103239604 Factor a perfect square on the left side: (d + 0.03773584906)(d + 0.03773584906) = 0.190103239604 Calculate the square root of the right side: 0.436008302 Break this problem into two subproblems by setting (d + 0.03773584906) equal to 0.436008302 and -0.436008302.Subproblem 1
d + 0.03773584906 = 0.436008302 Simplifying d + 0.03773584906 = 0.436008302 Reorder the terms: 0.03773584906 + d = 0.436008302 Solving 0.03773584906 + d = 0.436008302 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.03773584906' to each side of the equation. 0.03773584906 + -0.03773584906 + d = 0.436008302 + -0.03773584906 Combine like terms: 0.03773584906 + -0.03773584906 = 0.00000000000 0.00000000000 + d = 0.436008302 + -0.03773584906 d = 0.436008302 + -0.03773584906 Combine like terms: 0.436008302 + -0.03773584906 = 0.39827245294 d = 0.39827245294 Simplifying d = 0.39827245294Subproblem 2
d + 0.03773584906 = -0.436008302 Simplifying d + 0.03773584906 = -0.436008302 Reorder the terms: 0.03773584906 + d = -0.436008302 Solving 0.03773584906 + d = -0.436008302 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '-0.03773584906' to each side of the equation. 0.03773584906 + -0.03773584906 + d = -0.436008302 + -0.03773584906 Combine like terms: 0.03773584906 + -0.03773584906 = 0.00000000000 0.00000000000 + d = -0.436008302 + -0.03773584906 d = -0.436008302 + -0.03773584906 Combine like terms: -0.436008302 + -0.03773584906 = -0.47374415106 d = -0.47374415106 Simplifying d = -0.47374415106Solution
The solution to the problem is based on the solutions from the subproblems. d = {0.39827245294, -0.47374415106}
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