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Quizzes > Mathematics > Algebra

Only for Geniuses: Take the Ultimate Math Quiz

Ready for this test of genius? Prove your algebra skills now!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
paper art illustration with math symbols and numbers floating over coral background for a genius level quiz

This ultimate math quiz helps you measure algebra, number puzzles, and logic skills and see how far you can go. You'll get a score, spot gaps fast, and keep sharp with our related online quiz or a bit of genius trivia between rounds.

What is the value of 7 + 8 × 2?
23
46
30
15
According to the order of operations, multiplication is performed before addition. Here you first calculate 8 × 2 = 16 and then add 7 to get 23. Misordering these steps leads to incorrect results.
Simplify the expression: (x + 3) - (2x - 5).
-x - 2
3x + 2
-x + 8
-x + 2
Distribute the minus sign over (2x - 5) giving x + 3 - 2x + 5, then combine like terms: x - 2x = -x and 3 + 5 = 8. Thus the simplified form is -x + 8. This method applies to subtracting any parentheses expressions.
What is the greatest common divisor (GCD) of 24 and 36?
18
4
6
12
The GCD is the largest integer that divides both numbers. Factors of 24 are {1,2,3,4,6,8,12,24} and of 36 are {1,2,3,4,6,9,12,18,36}. The greatest common element is 12.
Solve for x: 3x = 12.
6
-4
4
3
Divide both sides of the equation by 3 to isolate x: x = 12 ÷ 3 = 4. This basic algebraic step removes the coefficient. Always perform the same operation on both sides of the equation.
What is the next prime number after 11?
17
12
13
15
A prime number is divisible only by 1 and itself. After 11, the next integer is 12 (composite), then 13, which has no divisors other than 1 and 13. Hence 13 is the next prime.
Evaluate: 5^2 - 3^2.
4
16
7
10
Compute each power first: 5^2 = 25 and 3^2 = 9. Subtract 9 from 25 to get 16. Always evaluate exponents before subtraction.
What is 50% of 80?
60
50
40
20
Fifty percent means one half of the value. Half of 80 is 40. Converting percentages to decimals (0.50 × 80) also yields 40.
If y = 2x and x = 5, what is y?
12
7
15
10
Substitute x = 5 into y = 2x to get y = 2 × 5 = 10. Function evaluation uses the current value of the variable. This is a direct substitution method. Function Evaluation
Solve for x: 2x + 5 = 17.
-6
6
11
7
Subtract 5 from both sides to get 2x = 12, then divide by 2 to find x = 6. Each operation preserves equality when applied to both sides.
Factor the expression: x^2 - 9.
(x - 3)(x + 3)
(x + 9)(x - 1)
(x - 9)(x + 1)
(x - 3)^2
This is a difference of squares: a^2 - b^2 = (a - b)(a + b). Here a = x and b = 3, giving (x - 3)(x + 3). Recognizing this pattern simplifies many algebraic expressions.
Solve for x: (x/3) + 4 = 11.
7
33
21
-7
Subtract 4 from both sides to get x/3 = 7, then multiply by 3 to find x = 21. Ensuring each step is applied equally maintains the solution.
Simplify the expression: 4(x - 2) + 6.
4x - 8
4x - 2
x - 2
4x + 2
First distribute 4 across (x - 2) to get 4x - 8, then add 6 yielding 4x - 2. Combining like terms produces the final result.
What is the least common multiple (LCM) of 4 and 6?
12
18
24
6
List multiples: 4 ? 4, 8, 12… and 6 ? 6, 12… The smallest common multiple is 12. LCM finds the smallest number divisible by both.
If f(x) = 2x^2 and x = 3, what is f(x)?
18
36
24
12
Substitute 3 into the function: f(3) = 2 × (3)^2 = 2 × 9 = 18. Functions map inputs to outputs via the given rule. Function Evaluation
Solve the system: x + y = 10 and x - y = 2.
x = 2, y = 8
x = 8, y = 2
x = 6, y = 4
x = 4, y = 6
Add the equations: (x + y) + (x - y) = 10 + 2 gives 2x = 12, so x = 6. Substitute back to find y = 4. This is the elimination method.
What is the slope of the line passing through (1, 2) and (3, 6)?
3/2
4/2
-2
2
Slope = (y2 - y1) / (x2 - x1) = (6 - 2) / (3 - 1) = 4 / 2 = 2. This ratio measures the steepness of the line.
Solve for x: x^2 - 5x + 6 = 0.
x = 2 or x = 3
x = 1 or x = 6
x = -2 or x = -3
x = 5 or x = 6
Factor the quadratic: (x - 2)(x - 3) = 0, so x = 2 or x = 3. Quadratic factoring finds roots efficiently.
What is the sum of the interior angles of a convex hexagon?
540°
720°
360°
900°
For any n-sided polygon, the sum is (n - 2) × 180°. Here n = 6, so (6 - 2) × 180° = 720°. This formula applies to all convex polygons.
Evaluate the expression: 3^3 × 2^4.
384
216
432
648
Compute powers separately: 3^3 = 27 and 2^4 = 16, then multiply: 27 × 16 = 432. Carrying out exponentiation before multiplication is key.
If a fair die is rolled twice, what is the probability of getting two even numbers?
1/3
1/4
1/2
1/8
Each roll has a 3/6 = 1/2 chance of being even. The events are independent, so multiply probabilities: (1/2) × (1/2) = 1/4. This is a basic rule of probability.
Simplify the rational expression: (2x^3 y^2) / (4x y).
x^2 y / 4
x^2 y / 2
x^3 y
x^2 y^2 / 2
Divide coefficients: 2/4 = 1/2. Subtract exponents: x^(3?1) = x^2, y^(2?1) = y. Thus result is (x^2 y)/2. Laws of exponents apply to both variables.
Find the determinant of the matrix [[2, 3], [1, 4]].
2
7
5
-5
For a 2×2 matrix [[a, b], [c, d]], the determinant is ad - bc. Here 2×4 - 3×1 = 8 - 3 = 5. Determinants measure area scaling or matrix invertibility.
Solve for x: sqrt(x + 3) = 5.
25
22
-22
2
Square both sides: x + 3 = 25, so x = 22. Check domain: x + 3 ? 0 holds when x = 22. Always verify solutions in the original equation.
How many 5-digit numbers have digits that sum to 5?
126
70
210
35
We count nonnegative solutions to d1 + d2 + d3 + d4 + d5 = 5 with 1 ? d1 ? 9. Let d1' = d1 - 1, then d1' + d2 + d3 + d4 + d5 = 4. The number of solutions is C(4 + 5 - 1, 5 - 1) = C(8,4) = 70. This uses the stars and bars method.
What is the general solution to the recurrence relation a? = 3a??? ? 2 with a? = 4?
3 × 2? + 1
3 × 3? + 1
3? + 2
4 × 3? ? 2
Solve the homogeneous part a?^h = C·3? and find a particular solution a?^p = 1. General solution: a? = C·3? + 1. Use a? = 4 to get C + 1 = 4, so C = 3. Hence a? = 3·3? + 1.
0
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Study Outcomes

  1. Analyze Complex Algebraic Expressions -

    Break down and simplify advanced equations to sharpen your skills in the algebra challenge quiz.

  2. Solve Numerical Puzzles Quickly -

    Apply strategic methods to tackle number-based challenges under time constraints in this tough maths quiz.

  3. Evaluate Your Math Genius Skills -

    Interpret your results from the math genius quiz to identify strengths and areas for improvement.

  4. Apply Logical Reasoning Techniques -

    Use pattern recognition and deduction to solve the most demanding problems in the test of genius.

  5. Benchmark Against Genius-Level Criteria -

    Compare your performance to only for genius standards to see if you meet the ultimate challenge threshold.

  6. Develop Advanced Problem-Solving Strategies -

    Refine and adapt methods for future algebra challenge quizzes and other high-level math tests.

Cheat Sheet

  1. Quadratic Equations & Vieta's Formulas -

    Master the standard form ax²+bx+c=0 and use Vieta's relations (sum of roots = - b/a, product = c/a) for lightning-fast solutions (MIT OpenCourseWare). For example, x² - 5x+6=0 gives roots 2 and 3 in one step. A handy mnemonic is "sum is negative b, product is c" to stay sharp in this only for genius level challenge.

  2. Arithmetic & Geometric Sequences -

    Review the nth-term formulas aₙ=a₝+(n - 1)d for arithmetic and gₙ=a₝·r❿❻¹ for geometric sequences (Khan Academy). Practice sum formulas Sₙ= n/2·(2a₝+(n - 1)d) and Sₙ=a₝(1 - r❿)/(1 - r) in quick drills. Mnemonic: "Add or multiply your way through" - perfect prep for any tough maths quiz.

  3. Modular Arithmetic & Congruences -

    Work with remainders using statements like 17≡2 mod 5 and solve puzzles via the Chinese Remainder Theorem (Art of Problem Solving). Treat mods like clock arithmetic to simplify huge exponent problems (e.g., 7¹❰❰ mod 5). This test of genius staple will boost your number theory confidence.

  4. Functional Equations & Symmetry Tricks -

    Spot patterns by plugging special values (x=0,1, - 1) and look for symmetry to guess linear or quadratic forms (USAMO problems archive). For f(x+1)=f(x)+2x+1, assume f(x)=ax²+bx+c and match coefficients. A playful tip: draw a quick table to see if f behaves like a parabola or a line.

  5. Factorization & Prime Magic -

    Sharpen skills with identities like a² - b²=(a - b)(a+b) and sum/difference of cubes: a³±b³=(a±b)(a²∓ab+b²) (Encyclopaedia of Mathematics). Break down large numbers into primes to solve puzzles on divisibility and LCM/GCD. Use the "SOAP" mnemonic (Same, Opposite, Always Positive) for sums/differences of odd powers.

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