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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1) Solve 3x^2 + 4x - 2 = 0 Step 1: Identify the coefficients. For the quadratic equation ax^2 + bx + c = 0, we have a=3, b=4, and c=-2. Step 2: Apply the quadratic formula. The quadratic formula is x = -b ± sqrt(b^2 - 4ac)2a. Substitute the values of a, b, and c: x = -(4) ± sqrt((4)^2 - 4(3)(-2))2(3) Step 3: Simplify the expression. x = -4 ± sqrt(16 - (-24))6 x = -4 ± sqrt(16 + 24)6 x = -4 ± sqrt(40)6 x = -4 ± sqrt(4 × 10)6 x = -4 ± 2sqrt(10)6 Divide all terms by 2: x = -2 ± sqrt(10)3 The two solutions are: x_1 = -2 + sqrt(10)3 x_2 = -2 - sqrt(10)3 The solutions are x = -2 ± sqrt(10)3. 2) Solve 2x^2 - 3x - 7 = 0 Step 1: Identify the coefficients. For ax^2 + bx + c = 0, we have a=2, b=-3, and c=-7. Step 2: Apply the quadratic formula. x = -b ± sqrt(b^2 - 4ac)2a Substitute the values: x = -(-3) ± sqrt((-3)^2 - 4(2)(-7))2(2) Step 3: Simplify the expression. x = 3 ± sqrt(9 - (-56))4 x = 3 ± sqrt(9 + 56)4 x = 3 ± sqrt(65)4 The two solutions are: x_1 = 3 + sqrt(65)4 x_2 = 3 - sqrt(65)4 The solutions are x = 3 ± sqrt(65)4. 3) Solve 3q^2 - 5q - 5 = 0 Step 1: Identify the coefficients. For aq^2 + bq + c = 0, we have a=3, b=-5, and c=-5. Step 2: Apply the quadratic formula. q = -b ± sqrt(b^2 - 4ac)2a Substitute the values: q = -(-5) ± sqrt((-5)^2 - 4(3)(-5))2(3) Step 3: Simplify the expression. q = 5 ± sqrt(25 - (-60))6 q = 5 ± sqrt(25 + 60)6 q = 5 ± sqrt(85)6 The two solutions are: q_1 = 5 + sqrt(85)6 q_2 = 5 - sqrt(85)6 The solutions are q = 5 ± sqrt(85)6. 4a) Show that x-3 = (5)/(3x) reduces to 3x^2 - 9x - 5 = 0 Step 1: Start with the given equation. x-3 = (5)/(3x) Step 2: Multiply both sides by 3x to eliminate the denominator. 3x(x-3) = 3x ((5)/(3x)) 3x^2 - 9x = 5 Step 3: Rearrange the terms to form a quadratic equation in standard form (ax^2 + bx + c = 0). Subtract 5 from both sides: 3x^2 - 9x - 5 = 0 This shows that the equation x-3 = (5)/(3x) reduces to 3x^2 - 9x - 5 = 0. 4b) Solve the equation 3x^2 - 9x - 5 = 0 Step 1: Identify the coefficients. For ax^2 + bx + c = 0, we have a=3, b=-9, and c=-5. Step 2: Apply the quadratic formula. x = -b ± sqrt(b^2 - 4ac)2a Substitute the values: x = -(-9) ± sqrt((-9)^2 - 4(3)(-5))2(3) Step 3: Simplify the expression. x = 9 ± sqrt(81 - (-60))6 x = 9 ± sqrt(81 + 60)6 x = 9 ± sqrt(141)6 The two solutions are: x_1 = 9 + sqrt(141)6 x_2 = 9 - sqrt(141)6 The solutions are x = 9 ± sqrt(141)6. Send me the next one 📸