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Simplifying 7v2 + -16 + -24v = 24 Reorder the terms: -16 + -24v + 7v2 = 24 Solving -16 + -24v + 7v2 = 24 Solving for variable 'v'. Reorder the terms: -16 + -24 + -24v + 7v2 = 24 + -24 Combine like terms: -16 + -24 = -40 -40 + -24v + 7v2 = 24 + -24 Combine like terms: 24 + -24 = 0 -40 + -24v + 7v2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -5.714285714 + -3.428571429v + v2 = 0 Move the constant term to the right: Add '5.714285714' to each side of the equation. -5.714285714 + -3.428571429v + 5.714285714 + v2 = 0 + 5.714285714 Reorder the terms: -5.714285714 + 5.714285714 + -3.428571429v + v2 = 0 + 5.714285714 Combine like terms: -5.714285714 + 5.714285714 = 0.000000000 0.000000000 + -3.428571429v + v2 = 0 + 5.714285714 -3.428571429v + v2 = 0 + 5.714285714 Combine like terms: 0 + 5.714285714 = 5.714285714 -3.428571429v + v2 = 5.714285714 The v term is -3.428571429v. Take half its coefficient (-1.714285715). Square it (2.938775513) and add it to both sides. Add '2.938775513' to each side of the equation. -3.428571429v + 2.938775513 + v2 = 5.714285714 + 2.938775513 Reorder the terms: 2.938775513 + -3.428571429v + v2 = 5.714285714 + 2.938775513 Combine like terms: 5.714285714 + 2.938775513 = 8.653061227 2.938775513 + -3.428571429v + v2 = 8.653061227 Factor a perfect square on the left side: (v + -1.714285715)(v + -1.714285715) = 8.653061227 Calculate the square root of the right side: 2.941608612 Break this problem into two subproblems by setting (v + -1.714285715) equal to 2.941608612 and -2.941608612.Subproblem 1
v + -1.714285715 = 2.941608612 Simplifying v + -1.714285715 = 2.941608612 Reorder the terms: -1.714285715 + v = 2.941608612 Solving -1.714285715 + v = 2.941608612 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1.714285715' to each side of the equation. -1.714285715 + 1.714285715 + v = 2.941608612 + 1.714285715 Combine like terms: -1.714285715 + 1.714285715 = 0.000000000 0.000000000 + v = 2.941608612 + 1.714285715 v = 2.941608612 + 1.714285715 Combine like terms: 2.941608612 + 1.714285715 = 4.655894327 v = 4.655894327 Simplifying v = 4.655894327Subproblem 2
v + -1.714285715 = -2.941608612 Simplifying v + -1.714285715 = -2.941608612 Reorder the terms: -1.714285715 + v = -2.941608612 Solving -1.714285715 + v = -2.941608612 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '1.714285715' to each side of the equation. -1.714285715 + 1.714285715 + v = -2.941608612 + 1.714285715 Combine like terms: -1.714285715 + 1.714285715 = 0.000000000 0.000000000 + v = -2.941608612 + 1.714285715 v = -2.941608612 + 1.714285715 Combine like terms: -2.941608612 + 1.714285715 = -1.227322897 v = -1.227322897 Simplifying v = -1.227322897Solution
The solution to the problem is based on the solutions from the subproblems. v = {4.655894327, -1.227322897}
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