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Simplifying 5a2 + -19a = 15 Reorder the terms: -19a + 5a2 = 15 Solving -19a + 5a2 = 15 Solving for variable 'a'. Reorder the terms: -15 + -19a + 5a2 = 15 + -15 Combine like terms: 15 + -15 = 0 -15 + -19a + 5a2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -3 + -3.8a + a2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + -3.8a + 3 + a2 = 0 + 3 Reorder the terms: -3 + 3 + -3.8a + a2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -3.8a + a2 = 0 + 3 -3.8a + a2 = 0 + 3 Combine like terms: 0 + 3 = 3 -3.8a + a2 = 3 The a term is -3.8a. Take half its coefficient (-1.9). Square it (3.61) and add it to both sides. Add '3.61' to each side of the equation. -3.8a + 3.61 + a2 = 3 + 3.61 Reorder the terms: 3.61 + -3.8a + a2 = 3 + 3.61 Combine like terms: 3 + 3.61 = 6.61 3.61 + -3.8a + a2 = 6.61 Factor a perfect square on the left side: (a + -1.9)(a + -1.9) = 6.61 Calculate the square root of the right side: 2.570992026 Break this problem into two subproblems by setting (a + -1.9) equal to 2.570992026 and -2.570992026.Subproblem 1
a + -1.9 = 2.570992026 Simplifying a + -1.9 = 2.570992026 Reorder the terms: -1.9 + a = 2.570992026 Solving -1.9 + a = 2.570992026 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.9' to each side of the equation. -1.9 + 1.9 + a = 2.570992026 + 1.9 Combine like terms: -1.9 + 1.9 = 0.0 0.0 + a = 2.570992026 + 1.9 a = 2.570992026 + 1.9 Combine like terms: 2.570992026 + 1.9 = 4.470992026 a = 4.470992026 Simplifying a = 4.470992026Subproblem 2
a + -1.9 = -2.570992026 Simplifying a + -1.9 = -2.570992026 Reorder the terms: -1.9 + a = -2.570992026 Solving -1.9 + a = -2.570992026 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.9' to each side of the equation. -1.9 + 1.9 + a = -2.570992026 + 1.9 Combine like terms: -1.9 + 1.9 = 0.0 0.0 + a = -2.570992026 + 1.9 a = -2.570992026 + 1.9 Combine like terms: -2.570992026 + 1.9 = -0.670992026 a = -0.670992026 Simplifying a = -0.670992026Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.470992026, -0.670992026}
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