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Simplifying 4a2 + 19a + -3 = 0 Reorder the terms: -3 + 19a + 4a2 = 0 Solving -3 + 19a + 4a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + 4.75a + a2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 4.75a + 0.75 + a2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 4.75a + a2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 4.75a + a2 = 0 + 0.75 4.75a + a2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 4.75a + a2 = 0.75 The a term is 4.75a. Take half its coefficient (2.375). Square it (5.640625) and add it to both sides. Add '5.640625' to each side of the equation. 4.75a + 5.640625 + a2 = 0.75 + 5.640625 Reorder the terms: 5.640625 + 4.75a + a2 = 0.75 + 5.640625 Combine like terms: 0.75 + 5.640625 = 6.390625 5.640625 + 4.75a + a2 = 6.390625 Factor a perfect square on the left side: (a + 2.375)(a + 2.375) = 6.390625 Calculate the square root of the right side: 2.527968552 Break this problem into two subproblems by setting (a + 2.375) equal to 2.527968552 and -2.527968552.Subproblem 1
a + 2.375 = 2.527968552 Simplifying a + 2.375 = 2.527968552 Reorder the terms: 2.375 + a = 2.527968552 Solving 2.375 + a = 2.527968552 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + a = 2.527968552 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + a = 2.527968552 + -2.375 a = 2.527968552 + -2.375 Combine like terms: 2.527968552 + -2.375 = 0.152968552 a = 0.152968552 Simplifying a = 0.152968552Subproblem 2
a + 2.375 = -2.527968552 Simplifying a + 2.375 = -2.527968552 Reorder the terms: 2.375 + a = -2.527968552 Solving 2.375 + a = -2.527968552 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.375' to each side of the equation. 2.375 + -2.375 + a = -2.527968552 + -2.375 Combine like terms: 2.375 + -2.375 = 0.000 0.000 + a = -2.527968552 + -2.375 a = -2.527968552 + -2.375 Combine like terms: -2.527968552 + -2.375 = -4.902968552 a = -4.902968552 Simplifying a = -4.902968552Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.152968552, -4.902968552}
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