Algebra Question

Grade Level: High School (Grades 9-12) or Introductory College
Time Limit: 90 minutes
Total Points: 100
Instructions:

• Answer all questions to the best of your ability.
• Show your work for full credit on calculation-based problems.
• Use a calculator only where permitted (noted in questions).
• This exam covers topics including basic operations, equations, inequalities, polynomials, quadratics, and functions.
• Good luck!

Section 1: Multiple Choice (20 questions, 2 points each, Total: 40 points)

Choose the best answer for each question.

1. Simplify: $3x+5x2x$.
   a) 6x
   b) 10x
   c) 6x – 2x
   d) 8x
2. Solve for x: $2x+3=7$.
   a) x = 2
   b) x = 4
   c) x = 5
   d) x = 10
3. What is the degree of the polynomial $4x^3+2x^{2}x+5$?
   a) 1
   b) 2
   c) 3
   d) 4
4. Factor: $x^{2}9$.
   a) (x – 3)(x + 3)
   b) (x – 9)(x + 1)
   c) (x^2 – 3)
   d) Cannot be factored
5. Solve: $x^2=16$.
   a) x = 4 only
   b) x = -4 only
   c) x = 4 or x = -4
   d) x = 16
6. Which inequality represents “x is greater than or equal to 5”?
   a) x > 5
   b) x 5
   c) x < 5
   d) x 5
7. Evaluate f(2) if f(x) = 3x + 4.
   a) 6
   b) 10
   c) 14
   d) 8
8. Simplify: $\frac{2x+4}{2}$.
   a) x + 2
   b) 2x + 2
   c) x + 4
   d) 2x
9. Solve the system: $y=2x+1$, $y=x+3$.
   a) (1, 3)
   b) (2, 5)
   c) (3, 7)
   d) No solution
10. What is the slope of the line y = 3x – 2?
    a) 3
    b) -2
    c) 1/3
    d) 2
11. Expand: (x + 2)^2.
    a) x^2 + 4
    b) x^2 + 2x + 4
    c) x^2 + 4x + 4
    d) x^2 + 2x + 2
12. Solve for x: $\frac{x}{3}=5$.
    a) x = 15
    b) x = 8
    c) x = 2
    d) x = 3
13. Which of the following is a quadratic equation?
    a) 2x + 3 = 0
    b) x^2 + 4x + 4 = 0
    c) x^3 + 1 = 0
    d) 5x = 10
14. Factor completely: $2x^2+4x$.
    a) 2x(x + 2)
    b) 2(x^2 + 2x)
    c) x(2x + 4)
    d) Cannot be factored
15. If a = 3 and b = 4, what is a^2 + b^2?
    a) 7
    b) 12
    c) 25
    d) 16
16. Solve: $3(x2)=9$.
    a) x = 5
    b) x = 3
    c) x = 6
    d) x = 2
17. What is the y-intercept of y = 2x + 3?
    a) 2
    b) 3
    c) 0
    d) 1
18. Simplify: $(2x{)}^2$.
    a) 2x^2
    b) 4x
    c) 4x^2
    d) 2x
19. Solve: $x^{2}4=0$.
    a) x = 2
    b) x = -2
    c) x = 2 or x = -2
    d) x = 4
20. Which function is linear?
    a) y = x^2
    b) y = 2x + 1
    c) y = 1/x
    d) y = x^3

Section 2: Short Answer (5 questions, 5 points each, Total: 25 points)

Provide brief answers, showing work where applicable.

1. Solve for x: $ 4x – 7 = 13 $.
2. Factor the polynomial: $ x^2 + 5x + 6 $.
3. Find the roots of the quadratic equation: $ x^2 – 6x + 9 = 0 $.
4. Simplify the expression: $ 2(x + 3) – 3(x – 1) $.
5. Determine if the point (2, 5) lies on the line y = 2x + 1. Explain.

Section 3: Long Answer (3 questions, 10 points each, Total: 30 points)

Show all work and explain your reasoning.

1. Solve the system of equations:
   $2x+y=5$
   $xy=1$
2. Graph the inequality: $ y > x + 2 $ on a coordinate plane. Describe the solution region.
3. A rectangle has a length that is 3 times its width. If the perimeter is 24 units, find the dimensions.

Section 4: Bonus (5 points)

1. Prove that the sum of the roots of the quadratic equation $a{x}^2+bx+c=0$ is $\frac{b}{a}$. (Hint: Use the quadratic formula.)

Answer Key

Section 1: Multiple Choice

1. a) 6x
2. a) x = 2
3. c) 3
4. a) (x – 3)(x + 3)
5. c) x = 4 or x = -4
6. b) x 5
7. b) 10
8. a) x + 2
9. b) (2, 5)
10. a) 3
11. c) x^2 + 4x + 4
12. a) x = 15
13. b) x^2 + 4x + 4 = 0
14. a) 2x(x + 2)
15. c) 25
16. a) x = 5
17. b) 3
18. c) 4x^2
19. c) x = 2 or x = -2
20. b) y = 2x + 1

Section 2: Short Answer
21. x = 5 (Add 7 to both sides: 4x = 20; divide by 4: x = 5)
22. (x + 2)(x + 3)
23. x = 3 (double root, since discriminant = 0)
24. 2x + 6 – 3x + 3 = -x + 9
25. Yes, because 2(2) + 1 = 5, which matches the y-coordinate.

Section 3: Long Answer
26. Add the equations: 3x = 6 x = 2; substitute: y = 5 – 4 = 1. Solution: (2, 1)
27. Shade above the line y = x + 2 (dashed line since >). Region is the half-plane above the line.
28. Let width = w, length = 3w. Perimeter: 2(3w + w) = 24 8w = 24 w = 3, length = 9.

Section 4: Bonus
29. Roots are $ frac{-b pm sqrt{b^2 – 4ac}}{2a} $. Sum: $ frac{-b + sqrt{…}}{2a} + frac{-b – sqrt{…}}{2a} = -frac{b}{a} $.

This exam is designed to test a range of algebra skills. If you’d like variations (e.g., more advanced topics, different formats, or solutions with full explanations), let me know!

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