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
Fresh day ~DÅRK, let's solve. We will solve each quadratic equation using the quadratic formula: x = -b ± sqrt(b^2 - 4ac)2a. 1) x^2 + 3x + 1 = 0 Step 1: Identify the coefficients. Here, a=1, b=3, c=1. Step 2: Substitute the values into the quadratic formula. x = -3 ± sqrt(3^2 - 4(1)(1))2(1) x = -3 ± sqrt(9 - 4)2 x = -3 ± sqrt(5)2 Step 3: Write the two solutions. x_1 = -3 + sqrt(5)2 x_2 = -3 - sqrt(5)2 The solutions are x = -3 ± sqrt(5)2. 2) x^2 - 7x - 2 = 0 Step 1: Identify the coefficients. Here, a=1, b=-7, c=-2. Step 2: Substitute the values into the quadratic formula. x = -(-7) ± sqrt((-7)^2 - 4(1)(-2))2(1) x = 7 ± sqrt(49 + 8)2 x = 7 ± sqrt(57)2 Step 3: Write the two solutions. x_1 = 7 + sqrt(57)2 x_2 = 7 - sqrt(57)2 The solutions are x = 7 ± sqrt(57)2. 3) 2m^2 + 10m - 5 = 0 Step 1: Identify the coefficients. Here, a=2, b=10, c=-5. Step 2: Substitute the values into the quadratic formula. m = -10 ± sqrt(10^2 - 4(2)(-5))2(2) m = -10 ± sqrt(100 + 40)4 m = -10 ± sqrt(140)4 Step 3: Simplify the radical sqrt(140). sqrt(140) = sqrt(4 × 35) = 2sqrt(35) Step 4: Substitute back and simplify the expression. m = -10 ± 2sqrt(35)4 m = 2(-5 ± sqrt(35))4 m = -5 ± sqrt(35)2 The solutions are m = -5 ± sqrt(35)2. 4) 5q^2 - q - 3 = 0 Step 1: Identify the coefficients. Here, a=5, b=-1, c=-3. Step 2: Substitute the values into the quadratic formula. q = -(-1) ± sqrt((-1)^2 - 4(5)(-3))2(5) q = 1 ± sqrt(1 + 60)10 q = 1 ± sqrt(61)10 Step 3: Write the two solutions. q_1 = 1 + sqrt(61)10 q_2 = 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 ax^2 + bx + c = 0. 10x^2 - 4x - 16 = 0 Divide by 2 to simplify: 5x^2 - 2x - 8 = 0 Now, identify the coefficients: a=5, b=-2, c=-8. Step 2: Substitute the values into the quadratic formula. x = -(-2) ± sqrt((-2)^2 - 4(5)(-8))2(5) x = 2 ± sqrt(4 + 160)10 x = 2 ± sqrt(164)10 Step 3: Simplify the radical sqrt(164). sqrt(164) = sqrt(4 × 41) = 2sqrt(41) Step 4: Substitute back and simplify the expression. x = 2 ± 2sqrt(41)10 x = 2(1 ± sqrt(41))10 x = 1 ± sqrt(41)5 The solutions are x = 1 ± sqrt(41)5. Send me the next one 📸