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Simplifying 7y(4y + -14) = 23 Reorder the terms: 7y(-14 + 4y) = 23 (-14 * 7y + 4y * 7y) = 23 (-98y + 28y2) = 23 Solving -98y + 28y2 = 23 Solving for variable 'y'. Reorder the terms: -23 + -98y + 28y2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + -98y + 28y2 = 0 Begin completing the square. Divide all terms by 28 the coefficient of the squared term: Divide each side by '28'. -0.8214285714 + -3.5y + y2 = 0 Move the constant term to the right: Add '0.8214285714' to each side of the equation. -0.8214285714 + -3.5y + 0.8214285714 + y2 = 0 + 0.8214285714 Reorder the terms: -0.8214285714 + 0.8214285714 + -3.5y + y2 = 0 + 0.8214285714 Combine like terms: -0.8214285714 + 0.8214285714 = 0.0000000000 0.0000000000 + -3.5y + y2 = 0 + 0.8214285714 -3.5y + y2 = 0 + 0.8214285714 Combine like terms: 0 + 0.8214285714 = 0.8214285714 -3.5y + y2 = 0.8214285714 The y term is -3.5y. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5y + 3.0625 + y2 = 0.8214285714 + 3.0625 Reorder the terms: 3.0625 + -3.5y + y2 = 0.8214285714 + 3.0625 Combine like terms: 0.8214285714 + 3.0625 = 3.8839285714 3.0625 + -3.5y + y2 = 3.8839285714 Factor a perfect square on the left side: (y + -1.75)(y + -1.75) = 3.8839285714 Calculate the square root of the right side: 1.970768523 Break this problem into two subproblems by setting (y + -1.75) equal to 1.970768523 and -1.970768523.Subproblem 1
y + -1.75 = 1.970768523 Simplifying y + -1.75 = 1.970768523 Reorder the terms: -1.75 + y = 1.970768523 Solving -1.75 + y = 1.970768523 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + y = 1.970768523 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + y = 1.970768523 + 1.75 y = 1.970768523 + 1.75 Combine like terms: 1.970768523 + 1.75 = 3.720768523 y = 3.720768523 Simplifying y = 3.720768523Subproblem 2
y + -1.75 = -1.970768523 Simplifying y + -1.75 = -1.970768523 Reorder the terms: -1.75 + y = -1.970768523 Solving -1.75 + y = -1.970768523 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + y = -1.970768523 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + y = -1.970768523 + 1.75 y = -1.970768523 + 1.75 Combine like terms: -1.970768523 + 1.75 = -0.220768523 y = -0.220768523 Simplifying y = -0.220768523Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.720768523, -0.220768523}
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