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Simplifying a(2a + -5) + 7(a + -3) = 0 Reorder the terms: a(-5 + 2a) + 7(a + -3) = 0 (-5 * a + 2a * a) + 7(a + -3) = 0 (-5a + 2a2) + 7(a + -3) = 0 Reorder the terms: -5a + 2a2 + 7(-3 + a) = 0 -5a + 2a2 + (-3 * 7 + a * 7) = 0 -5a + 2a2 + (-21 + 7a) = 0 Reorder the terms: -21 + -5a + 7a + 2a2 = 0 Combine like terms: -5a + 7a = 2a -21 + 2a + 2a2 = 0 Solving -21 + 2a + 2a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -10.5 + a + a2 = 0 Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + a + 10.5 + a2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + a + a2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + a + a2 = 0 + 10.5 a + a2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 a + a2 = 10.5 The a term is a. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + a2 = 10.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + a2 = 10.5 + 0.25 Combine like terms: 10.5 + 0.25 = 10.75 1.25 + a2 = 10.75 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 10.75 Calculate the square root of the right side: 3.278719262 Break this problem into two subproblems by setting (a + 0.5) equal to 3.278719262 and -3.278719262.Subproblem 1
a + 0.5 = 3.278719262 Simplifying a + 0.5 = 3.278719262 Reorder the terms: 0.5 + a = 3.278719262 Solving 0.5 + a = 3.278719262 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 3.278719262 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 3.278719262 + -0.5 a = 3.278719262 + -0.5 Combine like terms: 3.278719262 + -0.5 = 2.778719262 a = 2.778719262 Simplifying a = 2.778719262Subproblem 2
a + 0.5 = -3.278719262 Simplifying a + 0.5 = -3.278719262 Reorder the terms: 0.5 + a = -3.278719262 Solving 0.5 + a = -3.278719262 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -3.278719262 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -3.278719262 + -0.5 a = -3.278719262 + -0.5 Combine like terms: -3.278719262 + -0.5 = -3.778719262 a = -3.778719262 Simplifying a = -3.778719262Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.778719262, -3.778719262}
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