This geography problem is solved step by step below, with detailed explanations to help you understand the method and arrive at the correct answer.
ScanSolve AI Answer
To find the discriminant and describe the nature of the roots for each quadratic equation, we use the formula for the discriminant = b^2 - 4ac, where the quadratic equation is in the form ax^2 + bx + c = 0. The nature of the roots is determined as follows: If > 0, there are two distinct real roots. If = 0, there is one real root (a repeated root). If < 0, there are two complex conjugate roots (no real roots). --- a. x^2 - 4x + 3 = 0 Step 1: Identify a, b, c. Here, a = 1, b = -4, c = 3. Step 2: Calculate the discriminant. = b^2 - 4ac = (-4)^2 - 4(1)(3) = 16 - 12 = 4 Step 3: Describe the nature of the roots. Since = 4 > 0, the equation has two distinct real roots. The discriminant is 4. --- b. x^2 - 6x + 9 = 0 Step 1: Identify a, b, c. Here, a = 1, b = -6, c = 9. Step 2: Calculate the discriminant. = b^2 - 4ac = (-6)^2 - 4(1)(9) = 36 - 36 = 0 Step 3: Describe the nature of the roots. Since = 0, the equation has one real root (a repeated root). The discriminant is 0. --- c. x^2 + 2x + 5 = 0 Step 1: Identify a, b, c. Here, a = 1, b = 2, c = 5. Step 2: Calculate the discriminant. = b^2 - 4ac = (2)^2 - 4(1)(5) = 4 - 20 = -16 Step 3: Describe the nature of the roots. Since = -16 < 0, the equation has two complex conjugate roots (no real roots). The discriminant is -16. --- d. 3x^2 - 12x + 12 = 0 Step 1: Identify a, b, c. Here, a = 3, b = -12, c = 12. Step 2: Calculate the discriminant. = b^2 - 4ac = (-12)^2 - 4(3)(12) = 144 - 144 = 0 Step 3: Describe the nature of the roots. Since = 0, the equation has one real root (a repeated root). The discriminant is 0. --- e. x^2 - 7x + 10 = 0 Step 1: Identify a, b, c. Here, a = 1, b = -7, c = 10. Step 2: Calculate the discriminant. = b^2 - 4ac = (-7)^2 - 4(1)(10) = 49 - 40 = 9 Step 3: Describe the nature of the roots. Since = 9 > 0, the equation has two distinct real roots. The discriminant is 9.