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Simplifying (v + -9)(v + 8) = 6 Reorder the terms: (-9 + v)(v + 8) = 6 Reorder the terms: (-9 + v)(8 + v) = 6 Multiply (-9 + v) * (8 + v) (-9(8 + v) + v(8 + v)) = 6 ((8 * -9 + v * -9) + v(8 + v)) = 6 ((-72 + -9v) + v(8 + v)) = 6 (-72 + -9v + (8 * v + v * v)) = 6 (-72 + -9v + (8v + v2)) = 6 Combine like terms: -9v + 8v = -1v (-72 + -1v + v2) = 6 Solving -72 + -1v + v2 = 6 Solving for variable 'v'. Reorder the terms: -72 + -6 + -1v + v2 = 6 + -6 Combine like terms: -72 + -6 = -78 -78 + -1v + v2 = 6 + -6 Combine like terms: 6 + -6 = 0 -78 + -1v + v2 = 0 Begin completing the square. Move the constant term to the right: Add '78' to each side of the equation. -78 + -1v + 78 + v2 = 0 + 78 Reorder the terms: -78 + 78 + -1v + v2 = 0 + 78 Combine like terms: -78 + 78 = 0 0 + -1v + v2 = 0 + 78 -1v + v2 = 0 + 78 Combine like terms: 0 + 78 = 78 -1v + v2 = 78 The v term is -1v. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1v + 0.25 + v2 = 78 + 0.25 Reorder the terms: 0.25 + -1v + v2 = 78 + 0.25 Combine like terms: 78 + 0.25 = 78.25 0.25 + -1v + v2 = 78.25 Factor a perfect square on the left side: (v + -0.5)(v + -0.5) = 78.25 Calculate the square root of the right side: 8.845903006 Break this problem into two subproblems by setting (v + -0.5) equal to 8.845903006 and -8.845903006.Subproblem 1
v + -0.5 = 8.845903006 Simplifying v + -0.5 = 8.845903006 Reorder the terms: -0.5 + v = 8.845903006 Solving -0.5 + v = 8.845903006 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + v = 8.845903006 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + v = 8.845903006 + 0.5 v = 8.845903006 + 0.5 Combine like terms: 8.845903006 + 0.5 = 9.345903006 v = 9.345903006 Simplifying v = 9.345903006Subproblem 2
v + -0.5 = -8.845903006 Simplifying v + -0.5 = -8.845903006 Reorder the terms: -0.5 + v = -8.845903006 Solving -0.5 + v = -8.845903006 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + v = -8.845903006 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + v = -8.845903006 + 0.5 v = -8.845903006 + 0.5 Combine like terms: -8.845903006 + 0.5 = -8.345903006 v = -8.345903006 Simplifying v = -8.345903006Solution
The solution to the problem is based on the solutions from the subproblems. v = {9.345903006, -8.345903006}
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