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Simplifying 3.5(y2 + 4y + -5) + 2.4y2 + 4.6 = 0 Reorder the terms: 3.5(-5 + 4y + y2) + 2.4y2 + 4.6 = 0 (-5 * 3.5 + 4y * 3.5 + y2 * 3.5) + 2.4y2 + 4.6 = 0 (-17.5 + 14y + 3.5y2) + 2.4y2 + 4.6 = 0 Reorder the terms: -17.5 + 4.6 + 14y + 3.5y2 + 2.4y2 = 0 Combine like terms: -17.5 + 4.6 = -12.9 -12.9 + 14y + 3.5y2 + 2.4y2 = 0 Combine like terms: 3.5y2 + 2.4y2 = 5.9y2 -12.9 + 14y + 5.9y2 = 0 Solving -12.9 + 14y + 5.9y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 5.9 the coefficient of the squared term: Divide each side by '5.9'. -2.186440678 + 2.372881356y + y2 = 0 Move the constant term to the right: Add '2.186440678' to each side of the equation. -2.186440678 + 2.372881356y + 2.186440678 + y2 = 0 + 2.186440678 Reorder the terms: -2.186440678 + 2.186440678 + 2.372881356y + y2 = 0 + 2.186440678 Combine like terms: -2.186440678 + 2.186440678 = 0.000000000 0.000000000 + 2.372881356y + y2 = 0 + 2.186440678 2.372881356y + y2 = 0 + 2.186440678 Combine like terms: 0 + 2.186440678 = 2.186440678 2.372881356y + y2 = 2.186440678 The y term is 2.372881356y. Take half its coefficient (1.186440678). Square it (1.407641482) and add it to both sides. Add '1.407641482' to each side of the equation. 2.372881356y + 1.407641482 + y2 = 2.186440678 + 1.407641482 Reorder the terms: 1.407641482 + 2.372881356y + y2 = 2.186440678 + 1.407641482 Combine like terms: 2.186440678 + 1.407641482 = 3.59408216 1.407641482 + 2.372881356y + y2 = 3.59408216 Factor a perfect square on the left side: (y + 1.186440678)(y + 1.186440678) = 3.59408216 Calculate the square root of the right side: 1.895806467 Break this problem into two subproblems by setting (y + 1.186440678) equal to 1.895806467 and -1.895806467.Subproblem 1
y + 1.186440678 = 1.895806467 Simplifying y + 1.186440678 = 1.895806467 Reorder the terms: 1.186440678 + y = 1.895806467 Solving 1.186440678 + y = 1.895806467 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.186440678' to each side of the equation. 1.186440678 + -1.186440678 + y = 1.895806467 + -1.186440678 Combine like terms: 1.186440678 + -1.186440678 = 0.000000000 0.000000000 + y = 1.895806467 + -1.186440678 y = 1.895806467 + -1.186440678 Combine like terms: 1.895806467 + -1.186440678 = 0.709365789 y = 0.709365789 Simplifying y = 0.709365789Subproblem 2
y + 1.186440678 = -1.895806467 Simplifying y + 1.186440678 = -1.895806467 Reorder the terms: 1.186440678 + y = -1.895806467 Solving 1.186440678 + y = -1.895806467 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.186440678' to each side of the equation. 1.186440678 + -1.186440678 + y = -1.895806467 + -1.186440678 Combine like terms: 1.186440678 + -1.186440678 = 0.000000000 0.000000000 + y = -1.895806467 + -1.186440678 y = -1.895806467 + -1.186440678 Combine like terms: -1.895806467 + -1.186440678 = -3.082247145 y = -3.082247145 Simplifying y = -3.082247145Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.709365789, -3.082247145}
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