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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Morning Ridoskynation — let's get this done. To determine the nature of the roots of a quadratic equation ax^2 + bx + c = 0 without solving it, we use the discriminant, = b^2 - 4ac. • If > 0, the roots are real and distinct. • If = 0, the roots are real and equal. • If < 0, the roots are complex (non-real) and distinct. a) The given equation is 2x^2 + 2x = 10. Step 1: Rewrite the equation in standard quadratic form ax^2 + bx + c = 0. 2x^2 + 2x - 10 = 0 Step 2: Identify the coefficients a, b, c. Here, a=2, b=2, c=-10. Step 3: Calculate the discriminant = b^2 - 4ac. = (2)^2 - 4(2)(-10) = 4 - (-80) = 4 + 80 = 84 Step 4: Determine the nature of the roots based on the discriminant. Since = 84 > 0, the roots are real and distinct. b) The given equation is 2y - (14y)/(y+3) + 1 = 0. Step 1: Clear the denominator by multiplying the entire equation by (y+3), assuming y ≠ -3. 2y(y+3) - 14y + 1(y+3) = 0 2y^2 + 6y - 14y + y + 3 = 0 Step 2: Combine like terms to get the equation in standard quadratic form ay^2 + by + c = 0. 2y^2 - 7y + 3 = 0 Step 3: Identify the coefficients a, b, c. Here, a=2, b=-7, c=3. Step 4: Calculate the discriminant = b^2 - 4ac. = (-7)^2 - 4(2)(3) = 49 - 24 = 25 Step 5: Determine the nature of the roots based on the discriminant. Since = 25 > 0, the roots are real and distinct. c) The given equation is (2x+3)^2 + 12 = 0. Step 1: Expand the squared term (2x+3)^2. (2x+3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9 Step 2: Substitute the expanded term back into the equation and simplify to standard quadratic form ax^2 + bx + c = 0. 4x^2 + 12x + 9 + 12 = 0 4x^2 + 12x + 21 = 0 Step 3: Identify the coefficients a, b, c. Here, a=4, b=12, c=21. Step 4: Calculate the discriminant = b^2 - 4ac. = (12)^2 - 4(4)(21) = 144 - 16(21) = 144 - 336 = -192 Step 5: Determine the nature of the roots based on the discriminant. Since = -192 < 0, the roots are complex and distinct. d) The given equation is 6t^2 - 13t - 13 = 0. Step 1: The equation is already in standard quadratic form at^2 + bt + c = 0. Step 2: Identify the coefficients a, b, c. Here, a=6, b=-13, c=-13. Step 3: Calculate the discriminant = b^2 - 4ac. = (-13)^2 - 4(6)(-13) = 169 - (-312) = 169 + 312 = 481 Step 4: Determine the nature of the roots based on the discriminant. Since = 481 > 0, the roots are real and distinct. Send me the next one 📸