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.
ScanSolve AI Answer
Okay, ~DÅRK, let's solve these using the complete the square method again. 1) x^2 + 3x + 1 = 0 Step 1: Move the constant term to the right side. x^2 + 3x = -1 Step 2: Complete the square on the left side by adding ((3)/(2))^2 = (9)/(4) to both sides. x^2 + 3x + (9)/(4) = -1 + (9)/(4) Step 3: Factor the left side as a perfect square and simplify the right side. (x + (3)/(2))^2 = (-4)/(4) + (9)/(4) (x + (3)/(2))^2 = (5)/(4) Step 4: Take the square root of both sides. x + (3)/(2) = ±sqrt((5)/(4)) x + (3)/(2) = ±sqrt(5)2 Step 5: Solve for x. x = -(3)/(2) ± sqrt(5)2 x = -3 ± sqrt(5)2 The solutions are x = -3 ± sqrt(5)2. 2) x^2 - 7x - 2 = 0 Step 1: Move the constant term to the right side. x^2 - 7x = 2 Step 2: Complete the square on the left side by adding (-(7)/(2))^2 = (49)/(4) to both sides. x^2 - 7x + (49)/(4) = 2 + (49)/(4) Step 3: Factor the left side as a perfect square and simplify the right side. (x - (7)/(2))^2 = (8)/(4) + (49)/(4) (x - (7)/(2))^2 = (57)/(4) Step 4: Take the square root of both sides. x - (7)/(2) = ±sqrt((57)/(4)) x - (7)/(2) = ±sqrt(57)2 Step 5: Solve for x. x = (7)/(2) ± sqrt(57)2 x = 7 ± sqrt(57)2 The solutions are x = 7 ± sqrt(57)2. 3) 2m^2 + 10m - 5 = 0 Step 1: Divide the entire equation by the coefficient of m^2, which is 2. m^2 + 5m - (5)/(2) = 0 Step 2: Move the constant term to the right side. m^2 + 5m = (5)/(2) Step 3: Complete the square on the left side by adding ((5)/(2))^2 = (25)/(4) to both sides. m^2 + 5m + (25)/(4) = (5)/(2) + (25)/(4) Step 4: Factor the left side as a perfect square and simplify the right side. (m + (5)/(2))^2 = (10)/(4) + (25)/(4) (m + (5)/(2))^2 = (35)/(4) Step 5: Take the square root of both sides. m + (5)/(2) = ±sqrt((35)/(4)) m + (5)/(2) = ±sqrt(35)2 Step 6: Solve for m. m = -(5)/(2) ± sqrt(35)2 m = -5 ± sqrt(35)2 The solutions are m = -5 ± sqrt(35)2. 4) 5q^2 - q - 3 = 0 Step 1: Divide the entire equation by the coefficient of q^2, which is 5. q^2 - (1)/(5)q - (3)/(5) = 0 Step 2: Move the constant term to the right side. q^2 - (1)/(5)q = (3)/(5) Step 3: Complete the square on the left side by adding (-(1)/(10))^2 = (1)/(100) to both sides. q^2 - (1)/(5)q + (1)/(100) = (3)/(5) + (1)/(100) Step 4: Factor the left side as a perfect square and simplify the right side. (q - (1)/(10))^2 = (60)/(100) + (1)/(100) (q - (1)/(10))^2 = (61)/(100) Step 5: Take the square root of both sides. q - (1)/(10) = ±sqrt((61)/(100)) q - (1)/(10) = ±sqrt(61)10 Step 6: Solve for q. q = (1)/(10) ± sqrt(61)10 q = 1 ± sqrt(61)10 The solutions are q = 1 ± sqrt(61)10. 5) 10x^2 - 4x = 16 Step 1: Rewrite the equation in standard form and divide by the coefficient of x^2, which is 10. 10x^2 - 4x - 16 = 0 x^2 -