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Simplifying 5y2 + 3y2 + 10y + -2 = 0 Reorder the terms: -2 + 10y + 5y2 + 3y2 = 0 Combine like terms: 5y2 + 3y2 = 8y2 -2 + 10y + 8y2 = 0 Solving -2 + 10y + 8y2 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + 5y + 4y2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + 5y + 4y2)' equal to zero and attempt to solve: Simplifying -1 + 5y + 4y2 = 0 Solving -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 = 0.17539053 y = 0.17539053 Simplifying y = 0.17539053Subproblem 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 = -1.42539053 y = -1.42539053 Simplifying y = -1.42539053Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.17539053, -1.42539053}Solution
y = {0.17539053, -1.42539053}
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