This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Step 1: Introduce a substitution. Let y = a - x^2. The original equation (a - x^2)^2 = a + x can be rewritten as y^2 = a + x. Now we have a system of two equations: y = a - x^2 (1) y^2 = a + x (2) Step 2: Manipulate the system of equations. Subtract equation (1) from equation (2): y^2 - y = (a + x) - (a - x^2) y^2 - y = x + x^2 Rearrange the terms to one side: y^2 - x^2 - y - x = 0 Factor by grouping: (y - x)(y + x) - (y + x) = 0 (y + x)(y - x - 1) = 0 This gives two possible cases: Case A: y + x = 0 y = -x Case B: y - x - 1 = 0 y = x + 1 Step 3: Solve for x in Case A. Substitute y = -x into equation (1): -x = a - x^2 Rearrange into a quadratic equation: x^2 - x - a = 0 Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a: x = -(-1) ± sqrt((-1)^2 - 4(1)(-a))2(1) x = 1 ± sqrt(1 + 4a)2 For real solutions, 1 + 4a 0 a -(1)/(4). Step 4: Solve for x in Case B. Substitute y = x + 1 into equation (1): x + 1 = a - x^2 Rearrange into a quadratic equation: x^2 + x + 1 - a = 0 Use the quadratic formula: x = -1 ± sqrt(1^2 - 4(1)(1 - a))2(1) x = -1 ± sqrt(1 - 4 + 4a)2 x = -1 ± sqrt(4a - 3)2 For real solutions, 4a - 3 0 a (3)/(4). The possible values for x are: x = 1 ± sqrt(1 + 4a)2, x = -1 ± sqrt(4a - 3)2 Send me the next one 📸