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Simplifying 4p + 25 = 6(p + -3) * 3(4 + -3p) Reorder the terms: 25 + 4p = 6(p + -3) * 3(4 + -3p) Reorder the terms: 25 + 4p = 6(-3 + p) * 3(4 + -3p) Reorder the terms for easier multiplication: 25 + 4p = 6 * 3(-3 + p)(4 + -3p) Multiply 6 * 3 25 + 4p = 18(-3 + p)(4 + -3p) Multiply (-3 + p) * (4 + -3p) 25 + 4p = 18(-3(4 + -3p) + p(4 + -3p)) 25 + 4p = 18((4 * -3 + -3p * -3) + p(4 + -3p)) 25 + 4p = 18((-12 + 9p) + p(4 + -3p)) 25 + 4p = 18(-12 + 9p + (4 * p + -3p * p)) 25 + 4p = 18(-12 + 9p + (4p + -3p2)) Combine like terms: 9p + 4p = 13p 25 + 4p = 18(-12 + 13p + -3p2) 25 + 4p = (-12 * 18 + 13p * 18 + -3p2 * 18) 25 + 4p = (-216 + 234p + -54p2) Solving 25 + 4p = -216 + 234p + -54p2 Solving for variable 'p'. Reorder the terms: 25 + 216 + 4p + -234p + 54p2 = -216 + 234p + -54p2 + 216 + -234p + 54p2 Combine like terms: 25 + 216 = 241 241 + 4p + -234p + 54p2 = -216 + 234p + -54p2 + 216 + -234p + 54p2 Combine like terms: 4p + -234p = -230p 241 + -230p + 54p2 = -216 + 234p + -54p2 + 216 + -234p + 54p2 Reorder the terms: 241 + -230p + 54p2 = -216 + 216 + 234p + -234p + -54p2 + 54p2 Combine like terms: -216 + 216 = 0 241 + -230p + 54p2 = 0 + 234p + -234p + -54p2 + 54p2 241 + -230p + 54p2 = 234p + -234p + -54p2 + 54p2 Combine like terms: 234p + -234p = 0 241 + -230p + 54p2 = 0 + -54p2 + 54p2 241 + -230p + 54p2 = -54p2 + 54p2 Combine like terms: -54p2 + 54p2 = 0 241 + -230p + 54p2 = 0 Begin completing the square. Divide all terms by 54 the coefficient of the squared term: Divide each side by '54'. 4.462962963 + -4.259259259p + p2 = 0 Move the constant term to the right: Add '-4.462962963' to each side of the equation. 4.462962963 + -4.259259259p + -4.462962963 + p2 = 0 + -4.462962963 Reorder the terms: 4.462962963 + -4.462962963 + -4.259259259p + p2 = 0 + -4.462962963 Combine like terms: 4.462962963 + -4.462962963 = 0.000000000 0.000000000 + -4.259259259p + p2 = 0 + -4.462962963 -4.259259259p + p2 = 0 + -4.462962963 Combine like terms: 0 + -4.462962963 = -4.462962963 -4.259259259p + p2 = -4.462962963 The p term is -4.259259259p. Take half its coefficient (-2.12962963). Square it (4.535322361) and add it to both sides. Add '4.535322361' to each side of the equation. -4.259259259p + 4.535322361 + p2 = -4.462962963 + 4.535322361 Reorder the terms: 4.535322361 + -4.259259259p + p2 = -4.462962963 + 4.535322361 Combine like terms: -4.462962963 + 4.535322361 = 0.072359398 4.535322361 + -4.259259259p + p2 = 0.072359398 Factor a perfect square on the left side: (p + -2.12962963)(p + -2.12962963) = 0.072359398 Calculate the square root of the right side: 0.268997022 Break this problem into two subproblems by setting (p + -2.12962963) equal to 0.268997022 and -0.268997022.Subproblem 1
p + -2.12962963 = 0.268997022 Simplifying p + -2.12962963 = 0.268997022 Reorder the terms: -2.12962963 + p = 0.268997022 Solving -2.12962963 + p = 0.268997022 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.12962963' to each side of the equation. -2.12962963 + 2.12962963 + p = 0.268997022 + 2.12962963 Combine like terms: -2.12962963 + 2.12962963 = 0.00000000 0.00000000 + p = 0.268997022 + 2.12962963 p = 0.268997022 + 2.12962963 Combine like terms: 0.268997022 + 2.12962963 = 2.398626652 p = 2.398626652 Simplifying p = 2.398626652Subproblem 2
p + -2.12962963 = -0.268997022 Simplifying p + -2.12962963 = -0.268997022 Reorder the terms: -2.12962963 + p = -0.268997022 Solving -2.12962963 + p = -0.268997022 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.12962963' to each side of the equation. -2.12962963 + 2.12962963 + p = -0.268997022 + 2.12962963 Combine like terms: -2.12962963 + 2.12962963 = 0.00000000 0.00000000 + p = -0.268997022 + 2.12962963 p = -0.268997022 + 2.12962963 Combine like terms: -0.268997022 + 2.12962963 = 1.860632608 p = 1.860632608 Simplifying p = 1.860632608Solution
The solution to the problem is based on the solutions from the subproblems. p = {2.398626652, 1.860632608}
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