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Simplifying (4t + 16)(5t + -21) = 90 Reorder the terms: (16 + 4t)(5t + -21) = 90 Reorder the terms: (16 + 4t)(-21 + 5t) = 90 Multiply (16 + 4t) * (-21 + 5t) (16(-21 + 5t) + 4t * (-21 + 5t)) = 90 ((-21 * 16 + 5t * 16) + 4t * (-21 + 5t)) = 90 ((-336 + 80t) + 4t * (-21 + 5t)) = 90 (-336 + 80t + (-21 * 4t + 5t * 4t)) = 90 (-336 + 80t + (-84t + 20t2)) = 90 Combine like terms: 80t + -84t = -4t (-336 + -4t + 20t2) = 90 Solving -336 + -4t + 20t2 = 90 Solving for variable 't'. Reorder the terms: -336 + -90 + -4t + 20t2 = 90 + -90 Combine like terms: -336 + -90 = -426 -426 + -4t + 20t2 = 90 + -90 Combine like terms: 90 + -90 = 0 -426 + -4t + 20t2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-213 + -2t + 10t2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-213 + -2t + 10t2)' equal to zero and attempt to solve: Simplifying -213 + -2t + 10t2 = 0 Solving -213 + -2t + 10t2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -21.3 + -0.2t + t2 = 0.0 Move the constant term to the right: Add '21.3' to each side of the equation. -21.3 + -0.2t + 21.3 + t2 = 0.0 + 21.3 Reorder the terms: -21.3 + 21.3 + -0.2t + t2 = 0.0 + 21.3 Combine like terms: -21.3 + 21.3 = 0.0 0.0 + -0.2t + t2 = 0.0 + 21.3 -0.2t + t2 = 0.0 + 21.3 Combine like terms: 0.0 + 21.3 = 21.3 -0.2t + t2 = 21.3 The t term is -0.2t. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2t + 0.01 + t2 = 21.3 + 0.01 Reorder the terms: 0.01 + -0.2t + t2 = 21.3 + 0.01 Combine like terms: 21.3 + 0.01 = 21.31 0.01 + -0.2t + t2 = 21.31 Factor a perfect square on the left side: (t + -0.1)(t + -0.1) = 21.31 Calculate the square root of the right side: 4.616275555 Break this problem into two subproblems by setting (t + -0.1) equal to 4.616275555 and -4.616275555.Subproblem 1
t + -0.1 = 4.616275555 Simplifying t + -0.1 = 4.616275555 Reorder the terms: -0.1 + t = 4.616275555 Solving -0.1 + t = 4.616275555 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + t = 4.616275555 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + t = 4.616275555 + 0.1 t = 4.616275555 + 0.1 Combine like terms: 4.616275555 + 0.1 = 4.716275555 t = 4.716275555 Simplifying t = 4.716275555Subproblem 2
t + -0.1 = -4.616275555 Simplifying t + -0.1 = -4.616275555 Reorder the terms: -0.1 + t = -4.616275555 Solving -0.1 + t = -4.616275555 Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + t = -4.616275555 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + t = -4.616275555 + 0.1 t = -4.616275555 + 0.1 Combine like terms: -4.616275555 + 0.1 = -4.516275555 t = -4.516275555 Simplifying t = -4.516275555Solution
The solution to the problem is based on the solutions from the subproblems. t = {4.716275555, -4.516275555}Solution
t = {4.716275555, -4.516275555}
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