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Simplifying 3d + 10 = d(d + -4) Reorder the terms: 10 + 3d = d(d + -4) Reorder the terms: 10 + 3d = d(-4 + d) 10 + 3d = (-4 * d + d * d) 10 + 3d = (-4d + d2) Solving 10 + 3d = -4d + d2 Solving for variable 'd'. Combine like terms: 3d + 4d = 7d 10 + 7d + -1d2 = -4d + d2 + 4d + -1d2 Reorder the terms: 10 + 7d + -1d2 = -4d + 4d + d2 + -1d2 Combine like terms: -4d + 4d = 0 10 + 7d + -1d2 = 0 + d2 + -1d2 10 + 7d + -1d2 = d2 + -1d2 Combine like terms: d2 + -1d2 = 0 10 + 7d + -1d2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -10 + -7d + d2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + -7d + 10 + d2 = 0 + 10 Reorder the terms: -10 + 10 + -7d + d2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + -7d + d2 = 0 + 10 -7d + d2 = 0 + 10 Combine like terms: 0 + 10 = 10 -7d + d2 = 10 The d term is -7d. 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. -7d + 12.25 + d2 = 10 + 12.25 Reorder the terms: 12.25 + -7d + d2 = 10 + 12.25 Combine like terms: 10 + 12.25 = 22.25 12.25 + -7d + d2 = 22.25 Factor a perfect square on the left side: (d + -3.5)(d + -3.5) = 22.25 Calculate the square root of the right side: 4.716990566 Break this problem into two subproblems by setting (d + -3.5) equal to 4.716990566 and -4.716990566.Subproblem 1
d + -3.5 = 4.716990566 Simplifying d + -3.5 = 4.716990566 Reorder the terms: -3.5 + d = 4.716990566 Solving -3.5 + d = 4.716990566 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + d = 4.716990566 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + d = 4.716990566 + 3.5 d = 4.716990566 + 3.5 Combine like terms: 4.716990566 + 3.5 = 8.216990566 d = 8.216990566 Simplifying d = 8.216990566Subproblem 2
d + -3.5 = -4.716990566 Simplifying d + -3.5 = -4.716990566 Reorder the terms: -3.5 + d = -4.716990566 Solving -3.5 + d = -4.716990566 Solving for variable 'd'. Move all terms containing d to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + d = -4.716990566 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + d = -4.716990566 + 3.5 d = -4.716990566 + 3.5 Combine like terms: -4.716990566 + 3.5 = -1.216990566 d = -1.216990566 Simplifying d = -1.216990566Solution
The solution to the problem is based on the solutions from the subproblems. d = {8.216990566, -1.216990566}
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