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Simplifying 4y2 + -3y + 3 = 2y + 4 Reorder the terms: 3 + -3y + 4y2 = 2y + 4 Reorder the terms: 3 + -3y + 4y2 = 4 + 2y Solving 3 + -3y + 4y2 = 4 + 2y Solving for variable 'y'. Reorder the terms: 3 + -4 + -3y + -2y + 4y2 = 4 + 2y + -4 + -2y Combine like terms: 3 + -4 = -1 -1 + -3y + -2y + 4y2 = 4 + 2y + -4 + -2y Combine like terms: -3y + -2y = -5y -1 + -5y + 4y2 = 4 + 2y + -4 + -2y Reorder the terms: -1 + -5y + 4y2 = 4 + -4 + 2y + -2y Combine like terms: 4 + -4 = 0 -1 + -5y + 4y2 = 0 + 2y + -2y -1 + -5y + 4y2 = 2y + -2y Combine like terms: 2y + -2y = 0 -1 + -5y + 4y2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.25 + -1.25y + y2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -1.25y + 0.25 + y2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -1.25y + y2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -1.25y + y2 = 0 + 0.25 -1.25y + y2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -1.25y + y2 = 0.25 The y term is -1.25y. Take half its coefficient (-0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. -1.25y + 0.390625 + y2 = 0.25 + 0.390625 Reorder the terms: 0.390625 + -1.25y + y2 = 0.25 + 0.390625 Combine like terms: 0.25 + 0.390625 = 0.640625 0.390625 + -1.25y + y2 = 0.640625 Factor a perfect square on the left side: (y + -0.625)(y + -0.625) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (y + -0.625) equal to 0.80039053 and -0.80039053.Subproblem 1
y + -0.625 = 0.80039053 Simplifying y + -0.625 = 0.80039053 Reorder the terms: -0.625 + y = 0.80039053 Solving -0.625 + y = 0.80039053 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + y = 0.80039053 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + y = 0.80039053 + 0.625 y = 0.80039053 + 0.625 Combine like terms: 0.80039053 + 0.625 = 1.42539053 y = 1.42539053 Simplifying y = 1.42539053Subproblem 2
y + -0.625 = -0.80039053 Simplifying y + -0.625 = -0.80039053 Reorder the terms: -0.625 + y = -0.80039053 Solving -0.625 + y = -0.80039053 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + y = -0.80039053 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + y = -0.80039053 + 0.625 y = -0.80039053 + 0.625 Combine like terms: -0.80039053 + 0.625 = -0.17539053 y = -0.17539053 Simplifying y = -0.17539053Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.42539053, -0.17539053}
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