Solve in R 2x4 + 9x3 + 8x2 + 9x + 2 = 0

To solve the equation 2x^4 + 9x^3 + 8x^2 + 9x + 2 = 0 in R, we recognize it as a reciprocal equation because the coefficients are symmetric. Step 1: Check if x=0 is a solution. Substitute x=0 into the equation: 2(0)^4 + 9(0)^3 + 8(0)^2 + 9(0) + 2 = 2. Since 2 ≠ 0, x=0 is not a solution. Step 2: Divide the entire equation by x^2. Since x ≠ 0, we can divide by x^2: (2x^4)/(x^2) + (9x^3)/(x^2) + (8x^2)/(x^2) + (9x)/(x^2) + (2)/(x^2) = 0 2x^2 + 9x + 8 + (9)/(x) + (2)/(x^2) = 0 Step 3: Group terms with common coefficients. (2x^2 + (2)/(x^2)) + (9x + (9)/(x)) + 8 = 0 Factor out the common coefficients: 2(x^2 + (1)/(x^2)) + 9(x + (1)/(x)) + 8 = 0 Step 4: Make a substitution. Let y = x + (1)/(x). Then, square both sides: y^2 = (x + (1)/(x))^2 = x^2 + 2(x)((1)/(x)) + (1)/(x^2) = x^2 + 2 + (1)/(x^2). So, x^2 + (1)/(x^2) = y^2 - 2. Step 5: Substitute y into the equation. 2(y^2 - 2) + 9y + 8 = 0 2y^2 - 4 + 9y + 8 = 0 2y^2 + 9y + 4 = 0 Step 6: Solve the quadratic equation for y. We can factor this quadratic equation: 2y^2 + y + 8y + 4 = 0 y(2y + 1) + 4(2y + 1) = 0 (y + 4)(2y + 1) = 0 This gives two possible values for y: y + 4 = 0 y = -4 2y + 1 = 0 y = -(1)/(2) Step 7: Substitute back y = x + (1)/(x) and solve for x. Case 1: y = -4 x + (1)/(x) = -4 Multiply by x to clear the denominator: x^2 + 1 = -4x Rearrange into a standard quadratic form: x^2 + 4x + 1 = 0 Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a: x = -4 ± sqrt(4^2 - 4(1)(1))2(1) x = -4 ± sqrt(16 - 4)2 x = -4 ± sqrt(12)2 x = -4 ± 2sqrt(3)2 x = -2 ± sqrt(3) These are real solutions. Case 2: y = -(1)/(2) x + (1)/(x) = -(1)/(2) Multiply by 2x to clear the denominators: 2x^2 + 2 = -x Rearrange into a standard quadratic form: 2x^2 + x + 2 = 0 Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a: x = -1 ± sqrt(1^2 - 4(2)(2))2(2) x = -1 ± sqrt(1 - 16)4 x = -1 ± sqrt(-15)4 Since the discriminant is negative (sqrt(-15)), these solutions are complex numbers and are not in R. The real solutions are x = -2 + sqrt(3) and x = -2 - sqrt(3). The final answer is x = -2 ± sqrt(3). That's 2 down. 3 left today — send the next one.