Algebra and Equations

Let the roots of the quadratic equation x^2 - 12x + a = 0 be r and 3r, where one root is thrice the other. According to Vieta's formulas, the sum of the roots is equal to the coefficient of x with opposite sign, so r + 3r = 4r = 12, which gives r = 3. The product of the roots is equal to the constant term a, so r × 3r = 3 × 9 = 27. Therefore, the value of a is 27, which corresponds to option B.

Given the quadratic equation x² + 8x + 4 = 0, the sum of roots a + b = -8 and the product ab = 4. The expression a/b + b/a can be rewritten as (a² + b²)/ab. Using the identity a² + b² = (a + b)² - 2ab, we get ( (-8)² - 2*4 ) / 4 = (64 - 8) / 4 = 56 / 4 = 14.

The discriminant of the quadratic equation 3x² – 12x + 10 = 0 is Δ = (-12)² - 4*3*10 = 144 - 120 = 24, which is positive but not a perfect square. This means the roots are real, unequal, and irrational.

Solving the quadratic equation a² + 11a + 30 = 0 by factoring gives (a + 5)(a + 6) = 0, so the roots are a = -5 and a = -6. Solving the quadratic equation b² + 6b + 5 = 0 by factoring gives (b + 1)(b + 5) = 0, so the roots are b = -1 and b = -5. Comparing the roots, the largest root of a is -5 and the largest root of b is -1, so a < b. Therefore, the correct relationship is a < b, which corresponds to option A.

The quadratic equation x² + 2x - 15 = 0 can be factored as (x + 5)(x - 3) = 0. Setting each factor equal to zero gives the roots x = -5 and x = 3. Therefore, the correct roots are -5 and 3, which corresponds to option A.

Given the quadratic equation x² – 9x + 20 = 0, the sum of roots a + b = 9 and the product ab = 20. Using the identity a² + b² = (a + b)² - 2ab, we get a² + b² = 9² - 2×20 = 81 - 40 = 41. Therefore, a² + b² + ab = 41 + 20 = 61. Hence, the correct answer is option C.

The quadratic equation is 2x² + 3x - 9 = 0. Using the quadratic formula x = [-b ± sqrt(b² - 4ac)] / 2a with a=2, b=3, and c=-9, the discriminant is 81. The roots are x = ( -3 + 9 ) / 4 = 3/2 and x = ( -3 - 9 ) / 4 = -3. Therefore, the correct roots are 3/2 and -3, which corresponds to option B.

Given the quadratic equation 2x² + 5x + 3 = 0, the sum of roots is -5/2 and product of roots is 3/2. The quadratic with roots as reciprocals has sum = 1/(root1) + 1/(root2) = (root1 + root2)/(root1*root2) = (-5/2)/(3/2) = -5/3 and product = 1/(root1*root2) = 1/(3/2) = 2/3. So the new quadratic is x² - (sum)x + product = x² + (5/3)x + (2/3) = 0. Multiplying through by 3 gives 3x² + 5x + 2 = 0, which corresponds to option A after correcting the sign in the original options (see verification notes).

The quadratic equation 2x² + 5x + 2 = 0 factors as (2x + 1)(x + 2) = 0, giving roots x = -1/2 and x = -2, which corresponds to option A.

Solving the first quadratic equation 9a² + 18a + 5 = 0 using the quadratic formula gives roots a = -1/3 and a = -5/3 (approximately -0.333 and -1.667). Solving the second quadratic equation 2b² + 13b + 20 = 0 gives roots b = -5/2 and b = -4 (approximately -2.5 and -4). Comparing the roots, both roots of a are greater than the corresponding roots of b (since -0.333 > -2.5 and -1.667 > -4). Therefore, the correct comparison is a > b, which corresponds to option A.

Solving the first equation a^3 - 988 = 343 gives a^3 = 1331, so a = 11 (since 11^3 = 1331). Solving the second equation b^2 - 72 = 49 gives b^2 = 121, so b = ±11. Considering the positive root b = 11, we have a = b = 11. Therefore, the relation a ≥ b is true (since a = b). The option C (a < b) is incorrect. Given the options, the correct relation is B (a ≥ b).

Solving the first quadratic equation x² + 9x + 20 = 0 by factoring gives (x + 4)(x + 5) = 0, so the roots are x = -4 and x = -5. Solving the second quadratic equation y² + 5y + 6 = 0 by factoring gives (y + 2)(y + 3) = 0, so the roots are y = -2 and y = -3. Comparing the roots, both roots of x are less than both roots of y (since -5 < -3 and -4 < -2). Therefore, the correct relationship is x < y, which corresponds to option A.

The quadratic equation x² + x - 42 = 0 can be factored as (x + 7)(x - 6) = 0. Setting each factor equal to zero gives the roots x = -7 and x = 6. These roots correspond exactly to option D (-7, 6). Therefore, the correct answer is D, not A.

Solving the first quadratic equation a² - 13a + 42 = 0 by factoring gives roots a = 6 and a = 7. Solving the second quadratic equation b² - 15b + 56 = 0 by factoring gives roots b = 7 and b = 8. Comparing the roots, the possible values of a are 6 or 7, and the possible values of b are 7 or 8. Therefore, a can be less than or equal to b, but a > b is not always true. Hence, the correct inequality is a ≤ b, which corresponds to option D.

Solving the first equation a² + 8a + 16 = 0 can be rewritten as (a + 4)² = 0, which gives a single repeated root a = -4. Solving the second equation b² - 4b + 3 = 0 factors as (b - 1)(b - 3) = 0, giving two roots b = 1 and b = 3. Comparing the roots, a = -4 is less than both b = 1 and b = 3, so the relationship is a < b. Therefore, the correct answer is option A: If a < b.

The first quadratic equation x² + 3x - 18 = 0 factors as (x + 6)(x - 3) = 0, giving roots x = -6 and x = 3. The second quadratic equation y² + y - 30 = 0 factors as (y + 6)(y - 5) = 0, giving roots y = -6 and y = 5. Comparing the roots, both equations share the root -6, but the other roots are 3 for x and 5 for y. Since 3 ≤ 5 and -6 = -6, the relationship x ≤ y holds for the roots in general. Therefore, the correct answer is C (If x ≤ y).

Solving the first equation x² - x - 42 = 0 by factoring gives roots x = 7 or x = -6. Solving the second equation y² - 17y + 72 = 0 by factoring gives roots y = 8 or y = 9. Comparing the roots, the smallest x (-6) is less than the smallest y (8), and the largest x (7) is also less than the largest y (9). Therefore, for all roots, x < y is true. Hence, the correct relationship between x and y is x < y, which corresponds to option A.

The first quadratic equation x² + 5x + 6 = 0 factors as (x + 2)(x + 3) = 0, giving roots x = -2 and x = -3. The second quadratic equation y² + 9y + 14 = 0 factors as (y + 7)(y + 2) = 0, giving roots y = -7 and y = -2. Both equations share the root -2, but the other roots differ (-3 for x and -7 for y). Since the roots overlap and the other roots do not have a consistent inequality relationship, the relationship between x and y cannot be definitively established. Therefore, the correct answer is D: "If x = y or the relationship between x and y cannot be established."

Solving the first quadratic equation x² + 11x + 30 = 0 gives roots x = -5 and x = -6. Solving the second quadratic y² + 15y + 56 = 0 gives roots y = -7 and y = -8. Since the largest root of x (-5) is greater than the largest root of y (-7), it follows that x > y, making option B correct.

The quadratic equation 3x² – 7x – 6 = 0 can be solved using the quadratic formula or factoring. Factoring gives (3x + 2)(x - 3) = 0, so the roots are x = -2/3 and x = 3, which corresponds to option B.

The equation is 21345 * 45 = 4270 * x. Calculating the left side: 21345 * 45 = 960,525. To find x, divide 960,525 by 4270, which gives approximately 224.95. Among the options, 225 (option C) is the closest and correct answer.

Let the number of children be x. Each child gets x/8 notebooks, so total notebooks = x * (x/8) = x^2/8. If the number of children is halved (x/2), each gets 16 notebooks, so total notebooks = (x/2) * 16 = 8x. Equate total notebooks: x^2/8 = 8x, which simplifies to x^2 = 64x, so x = 64 (ignoring zero). Total notebooks = 64^2/8 = 512.

Solving the equation for the unknown denominator, substituting each option shows that when ? = 2000, the equation balances to 1680, confirming A as correct.

The equation is ∛(x/9261) = 11/21. Cubing both sides gives x/9261 = (11/21)^3 = 1331/9261. Multiplying both sides by 9261 yields x = 1331. However, the question is written as ∛?/9261 = 11/21, which implies the cube root of (? divided by 9261) equals 11/21, so the value inside the cube root is ?/9261 = (11/21)^3 = 1331/9261. Therefore, ? = 1331. The correct answer is B: 1331.

Simplify the left side first: (98 / 7) = 14, then 14 / 2 = 7. So the equation becomes 7 / ? = 14. Multiply both sides by ?: 7 = 14 × ?. Divide both sides by 14: ? = 7 / 14 = 1/2. Therefore, the correct answer is A (1/2). The original correct_answer field was B, which is incorrect. The explanation and calculation confirm that the correct answer is A.