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Simplifying 25 + -16v * 2 = 12v(v + 5) Reorder the terms for easier multiplication: 25 + -16 * 2v = 12v(v + 5) Multiply -16 * 2 25 + -32v = 12v(v + 5) Reorder the terms: 25 + -32v = 12v(5 + v) 25 + -32v = (5 * 12v + v * 12v) 25 + -32v = (60v + 12v2) Solving 25 + -32v = 60v + 12v2 Solving for variable 'v'. Combine like terms: -32v + -60v = -92v 25 + -92v + -12v2 = 60v + 12v2 + -60v + -12v2 Reorder the terms: 25 + -92v + -12v2 = 60v + -60v + 12v2 + -12v2 Combine like terms: 60v + -60v = 0 25 + -92v + -12v2 = 0 + 12v2 + -12v2 25 + -92v + -12v2 = 12v2 + -12v2 Combine like terms: 12v2 + -12v2 = 0 25 + -92v + -12v2 = 0 Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. -2.083333333 + 7.666666667v + v2 = 0 Move the constant term to the right: Add '2.083333333' to each side of the equation. -2.083333333 + 7.666666667v + 2.083333333 + v2 = 0 + 2.083333333 Reorder the terms: -2.083333333 + 2.083333333 + 7.666666667v + v2 = 0 + 2.083333333 Combine like terms: -2.083333333 + 2.083333333 = 0.000000000 0.000000000 + 7.666666667v + v2 = 0 + 2.083333333 7.666666667v + v2 = 0 + 2.083333333 Combine like terms: 0 + 2.083333333 = 2.083333333 7.666666667v + v2 = 2.083333333 The v term is 7.666666667v. Take half its coefficient (3.833333334). Square it (14.69444445) and add it to both sides. Add '14.69444445' to each side of the equation. 7.666666667v + 14.69444445 + v2 = 2.083333333 + 14.69444445 Reorder the terms: 14.69444445 + 7.666666667v + v2 = 2.083333333 + 14.69444445 Combine like terms: 2.083333333 + 14.69444445 = 16.777777783 14.69444445 + 7.666666667v + v2 = 16.777777783 Factor a perfect square on the left side: (v + 3.833333334)(v + 3.833333334) = 16.777777783 Calculate the square root of the right side: 4.096068576 Break this problem into two subproblems by setting (v + 3.833333334) equal to 4.096068576 and -4.096068576.Subproblem 1
v + 3.833333334 = 4.096068576 Simplifying v + 3.833333334 = 4.096068576 Reorder the terms: 3.833333334 + v = 4.096068576 Solving 3.833333334 + v = 4.096068576 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-3.833333334' to each side of the equation. 3.833333334 + -3.833333334 + v = 4.096068576 + -3.833333334 Combine like terms: 3.833333334 + -3.833333334 = 0.000000000 0.000000000 + v = 4.096068576 + -3.833333334 v = 4.096068576 + -3.833333334 Combine like terms: 4.096068576 + -3.833333334 = 0.262735242 v = 0.262735242 Simplifying v = 0.262735242Subproblem 2
v + 3.833333334 = -4.096068576 Simplifying v + 3.833333334 = -4.096068576 Reorder the terms: 3.833333334 + v = -4.096068576 Solving 3.833333334 + v = -4.096068576 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-3.833333334' to each side of the equation. 3.833333334 + -3.833333334 + v = -4.096068576 + -3.833333334 Combine like terms: 3.833333334 + -3.833333334 = 0.000000000 0.000000000 + v = -4.096068576 + -3.833333334 v = -4.096068576 + -3.833333334 Combine like terms: -4.096068576 + -3.833333334 = -7.92940191 v = -7.92940191 Simplifying v = -7.92940191Solution
The solution to the problem is based on the solutions from the subproblems. v = {0.262735242, -7.92940191}
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