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Simplifying -1 = 4y2 + -7y + 1 Reorder the terms: -1 = 1 + -7y + 4y2 Solving -1 = 1 + -7y + 4y2 Solving for variable 'y'. Combine like terms: -1 + -1 = -2 -2 + 7y + -4y2 = 1 + -7y + 4y2 + -1 + 7y + -4y2 Reorder the terms: -2 + 7y + -4y2 = 1 + -1 + -7y + 7y + 4y2 + -4y2 Combine like terms: 1 + -1 = 0 -2 + 7y + -4y2 = 0 + -7y + 7y + 4y2 + -4y2 -2 + 7y + -4y2 = -7y + 7y + 4y2 + -4y2 Combine like terms: -7y + 7y = 0 -2 + 7y + -4y2 = 0 + 4y2 + -4y2 -2 + 7y + -4y2 = 4y2 + -4y2 Combine like terms: 4y2 + -4y2 = 0 -2 + 7y + -4y2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 0.5 + -1.75y + y2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -1.75y + -0.5 + y2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -1.75y + y2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -1.75y + y2 = 0 + -0.5 -1.75y + y2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -1.75y + y2 = -0.5 The y term is -1.75y. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75y + 0.765625 + y2 = -0.5 + 0.765625 Reorder the terms: 0.765625 + -1.75y + y2 = -0.5 + 0.765625 Combine like terms: -0.5 + 0.765625 = 0.265625 0.765625 + -1.75y + y2 = 0.265625 Factor a perfect square on the left side: (y + -0.875)(y + -0.875) = 0.265625 Calculate the square root of the right side: 0.515388203 Break this problem into two subproblems by setting (y + -0.875) equal to 0.515388203 and -0.515388203.Subproblem 1
y + -0.875 = 0.515388203 Simplifying y + -0.875 = 0.515388203 Reorder the terms: -0.875 + y = 0.515388203 Solving -0.875 + y = 0.515388203 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + y = 0.515388203 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + y = 0.515388203 + 0.875 y = 0.515388203 + 0.875 Combine like terms: 0.515388203 + 0.875 = 1.390388203 y = 1.390388203 Simplifying y = 1.390388203Subproblem 2
y + -0.875 = -0.515388203 Simplifying y + -0.875 = -0.515388203 Reorder the terms: -0.875 + y = -0.515388203 Solving -0.875 + y = -0.515388203 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + y = -0.515388203 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + y = -0.515388203 + 0.875 y = -0.515388203 + 0.875 Combine like terms: -0.515388203 + 0.875 = 0.359611797 y = 0.359611797 Simplifying y = 0.359611797Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.390388203, 0.359611797}
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