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Simplifying -4t(5t + -5) + 8t = 7t + -4 Reorder the terms: -4t(-5 + 5t) + 8t = 7t + -4 (-5 * -4t + 5t * -4t) + 8t = 7t + -4 (20t + -20t2) + 8t = 7t + -4 Reorder the terms: 20t + 8t + -20t2 = 7t + -4 Combine like terms: 20t + 8t = 28t 28t + -20t2 = 7t + -4 Reorder the terms: 28t + -20t2 = -4 + 7t Solving 28t + -20t2 = -4 + 7t Solving for variable 't'. Reorder the terms: 4 + 28t + -7t + -20t2 = -4 + 7t + 4 + -7t Combine like terms: 28t + -7t = 21t 4 + 21t + -20t2 = -4 + 7t + 4 + -7t Reorder the terms: 4 + 21t + -20t2 = -4 + 4 + 7t + -7t Combine like terms: -4 + 4 = 0 4 + 21t + -20t2 = 0 + 7t + -7t 4 + 21t + -20t2 = 7t + -7t Combine like terms: 7t + -7t = 0 4 + 21t + -20t2 = 0 Begin completing the square. Divide all terms by -20 the coefficient of the squared term: Divide each side by '-20'. -0.2 + -1.05t + t2 = 0.0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + -1.05t + 0.2 + t2 = 0.0 + 0.2 Reorder the terms: -0.2 + 0.2 + -1.05t + t2 = 0.0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + -1.05t + t2 = 0.0 + 0.2 -1.05t + t2 = 0.0 + 0.2 Combine like terms: 0.0 + 0.2 = 0.2 -1.05t + t2 = 0.2 The t term is -1.05t. Take half its coefficient (-0.525). Square it (0.275625) and add it to both sides. Add '0.275625' to each side of the equation. -1.05t + 0.275625 + t2 = 0.2 + 0.275625 Reorder the terms: 0.275625 + -1.05t + t2 = 0.2 + 0.275625 Combine like terms: 0.2 + 0.275625 = 0.475625 0.275625 + -1.05t + t2 = 0.475625 Factor a perfect square on the left side: (t + -0.525)(t + -0.525) = 0.475625 Calculate the square root of the right side: 0.689655711 Break this problem into two subproblems by setting (t + -0.525) equal to 0.689655711 and -0.689655711.Subproblem 1
t + -0.525 = 0.689655711 Simplifying t + -0.525 = 0.689655711 Reorder the terms: -0.525 + t = 0.689655711 Solving -0.525 + t = 0.689655711 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.525' to each side of the equation. -0.525 + 0.525 + t = 0.689655711 + 0.525 Combine like terms: -0.525 + 0.525 = 0.000 0.000 + t = 0.689655711 + 0.525 t = 0.689655711 + 0.525 Combine like terms: 0.689655711 + 0.525 = 1.214655711 t = 1.214655711 Simplifying t = 1.214655711Subproblem 2
t + -0.525 = -0.689655711 Simplifying t + -0.525 = -0.689655711 Reorder the terms: -0.525 + t = -0.689655711 Solving -0.525 + t = -0.689655711 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.525' to each side of the equation. -0.525 + 0.525 + t = -0.689655711 + 0.525 Combine like terms: -0.525 + 0.525 = 0.000 0.000 + t = -0.689655711 + 0.525 t = -0.689655711 + 0.525 Combine like terms: -0.689655711 + 0.525 = -0.164655711 t = -0.164655711 Simplifying t = -0.164655711Solution
The solution to the problem is based on the solutions from the subproblems. t = {1.214655711, -0.164655711}
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