curiouskids

Due: Aug 7th

Questions

Comprehension

  1. A quadratic equation has roots at x = 3 and x = -4. What is the equation in standard form?
  2. The product of two consecutive positive even numbers is 528. What are the numbers?
  3. The sum of the squares of two consecutive positive integers is 85. What are the integers?
  4. A rectangle has an area of 48 square units and a length that is three times its width. What are the dimensions of the rectangle?
  5. If the complex number z = 3 + 4i, what is the real part of z? What is the imaginary part of z?
  6. Solve the system of equations: 3x - 2y = 7 and 5x + y = 26.
  7. The seventh term in an arithmetic sequence is 31, and the eleventh term is 47. What is the common difference?
  8. The second term in a geometric sequence is 3, and the fifth term is 81. What is the common ratio?
  9. What is the sum of the interior angles of a regular hexagon?
  10. You are choosing a password for an account. The password must be exactly six characters long, and it can contain any of the 26 letters of the alphabet (uppercase or lowercase) and the numbers 0 through 9. How many different passwords are possible?
  11. You invested $40,000 between two accounts. The first account pays 3.5% annual interest compounded quarterly and the second pays 4% annual interest compounded annually. After one year, the total interest from both accounts is $1,550. How much money did you put in each account?
  12. Find the area of a trapezoid with bases of 8 and 12 units and a height of 5 units.

Solutions

  1. The roots of the quadratic equation are x = 3 and x = -4. Therefore, the equation can be expressed in standard form as (x - 3)(x + 4) = 0, which simplifies to x^2 + x - 12 = 0.

  2. Let’s denote the two consecutive even numbers as x and x + 2. We know that x * (x + 2) = 528. Solving this equation gives x = 22 and x + 2 = 24. Therefore, the numbers are 22 and 24.

  3. Let’s denote the two consecutive integers as x and x + 1. We know that x^2 + (x + 1)^2 = 85. Solving this equation gives x = 6 and x + 1 = 7. Therefore, the integers are 6 and 7.

  4. Let’s denote the width as x and the length as 3x. The area of the rectangle is x * 3x = 48, which gives x = 4 and 3x = 12. Therefore, the dimensions of the rectangle are 4 units by 12 units.

  5. The complex number z = 3 + 4i has a real part of 3 and an imaginary part of 4.

  6. The system of equations 3x - 2y = 7 5x + y = 26 can be solved using substitution or elimination. A possible solution is x = 5, y = -1.

  7. The common difference in an arithmetic sequence can be found by subtracting the seventh term from the eleventh term and dividing by 4. Thus, d = (47 - 31) / 4 = 4.

  8. The common ratio in a geometric sequence can be found by taking the fifth term divided by the second term and then taking the cubed root (because they are 3 terms apart). Thus, r = cuberoot(81 / 3) = 3.

  9. The sum of the interior angles of any n-gon is (n - 2) * 180 degrees. For a hexagon (n = 6), this is 720 degrees.

  10. The number of possible passwords is the total number of characters (52 letters + 10 numbers) raised to the power of the number of characters in the password. So, there are 62^6 = 56,800,235,584 possible passwords.

  11. This question involves solving for two variables with compound interest. It might be easier to use a financial calculator or a software tool that can handle compound interest calculations to solve this problem.

  12. The area of a trapezoid is given by 1/2 * (base1 + base2) * height = 1/2 * (8 + 12) * 5 = 50 square units.

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