Algebra Quiz: Identities, Surds and Linear Equations Practice

Spoilers! These are the Starter quiz’s answers. Easy, Medium, Hard and Mixed rounds show their own answers when you finish.

1. If x + 1/x = 5, what is the value of x² + 1/x²?

   • 25
   • 23
   • 27
   • 21

   Answer: 23. Square both sides: x² + 2 + 1/x² = 25, so x² + 1/x² = 23.

2. If a + b = 10 and ab = 21, what is the value of a³ + b³?

   • 370
   • 390
   • 350
   • 430

   Answer: 370. a³ + b³ = (a + b)³ − 3ab(a + b) = 1000 − 630 = 370. (Check: a, b = 3 and 7: 27 + 343 = 370.)

3. If x − 1/x = 3, what is the value of x³ − 1/x³?

   • 27
   • 30
   • 36
   • 33

   Answer: 36. x³ − 1/x³ = k³ + 3k when x − 1/x = k: 27 + 9 = 36.

4. If a = 5, b = −2 and c = −3, what is the value of a³ + b³ + c³?

   • 30
   • -90
   • 60
   • 90

   Answer: 90. Since a + b + c = 0, a³ + b³ + c³ = 3abc = 3 × 5 × (−2) × (−3) = 90.

5. Solve for x: 3(x − 2) − 2(x − 5) = 9.

   • 5
   • 3
   • 7
   • 4

   Answer: 5. Expand: 3x − 6 − 2x + 10 = 9, so x + 4 = 9 and x = 5.

6. If 2x + 3y = 12 and 3x − y = 7, what is the value of x + y?

   • 4
   • 6
   • 5
   • 7

   Answer: 5. From the second equation y = 3x − 7. Then 2x + 9x − 21 = 12, so x = 3 and y = 2: x + y = 5.

7. What is the area of the triangle formed by the line 3x + 4y = 12 and the two coordinate axes?

   • 12 sq units
   • 7 sq units
   • 8 sq units
   • 6 sq units

   Answer: 6 sq units. The line meets the axes at (4, 0) and (0, 3). Area = 1/2 × 4 × 3 = 6 square units.

8. Simplify: sqrt(50) + sqrt(18) − sqrt(8)

   • 4 sqrt(2)
   • 6 sqrt(2)
   • 8 sqrt(2)
   • 5 sqrt(2)

   Answer: 6 sqrt(2). Pull out squares: 5 sqrt(2) + 3 sqrt(2) − 2 sqrt(2) = 6 sqrt(2). Only like surds can be added.

9. If 9^x = 27^(x − 1), what is x?

   • 2
   • 4
   • 3
   • 6

   Answer: 3. Use base 3: 3^(2x) = 3^(3x − 3). Equate powers: 2x = 3x − 3, so x = 3.

10. 1/(sqrt(5) − 2) is equal to

    • sqrt(5) + 2
    • sqrt(5) − 2
    • 2 − sqrt(5)
    • (sqrt(5) + 2)/3

    Answer: sqrt(5) + 2. Multiply top and bottom by the conjugate sqrt(5) + 2: the denominator becomes 5 − 4 = 1.