🏅 Top 10 in the last 30 days
No results yet. Be the first!
Test
Algebraic identities and factorisation
Dynamic tasks on square identities, difference of squares, common factors and polynomial factorisation.
Questions: 10Estimated time: 21 minutes
Start a test→
Enable solution history?
Test results will be stored only on this device in your browser’s local storage. They will not be sent to the Catestia.com database. If you disable this feature, all saved history will be permanently deleted.
Disable and delete history?
Disabling solution history will permanently delete all test results saved on this device. Are you sure you want to continue?
Preparing the test…
Question 1out of 10
What is the sum of all solutions of 3x² − 15x = 0?
Factor out x. A product is zero when at least one factor is zero. Find both solutions and add them.
Question 2out of 10
A student wrote: (x + 6)2 = x2 + 62. What is the error?
Square of a sum: (x + a)² = x² + 2ax + a². The middle term comes from adding the two cross products ax. Thus the x coefficient is 2a; x² + a² alone is incomplete.
Question 3out of 10
The solutions of 3x² = 432 are marked on a number line. What is the distance between the two points?
Divide by the coefficient of x². Find both solutions with opposite signs. The distance is the larger solution minus the smaller one.
Question 4out of 10
The number 2 solves 3x² + c = 0. Find c.
Substitute the given solution for x. Square it first, multiply by the coefficient, then isolate c in the resulting equality.
Question 5out of 10
Factorise: x2 − 22.
A difference of squares factors as a² − b² = (a − b)(a + b). Apply this to the given bases. Expanding the product cancels the cross terms.
Question 6out of 10
In (x + 2)2 = x2 + kx + 4, find coefficient k.
Square of a sum: (x + a)² = x² + 2ax + a². The middle term comes from adding the two cross products ax. Thus the x coefficient is 2a; x² + a² alone is incomplete.
Question 7out of 10
Factorise: x2 + 6x + 9.
Find two numbers whose sum is s and product is p. If they are a and b, then x² + sx + p = (x + a)(x + b). Check both conditions.
Question 8out of 10
Factor out the common factor: 5x + 5·6.
Find a factor common to all terms. Divide each term by it and write the quotients in parentheses: ax + ab = a(x + b). Check by expanding.
Question 9out of 10
A square plate loses 50 cm² of material and has 50 cm² left. Its original side length x satisfies x² − 50 = 50. Find the original side length.
Add the removed area to the remaining area. Solve x² = S. Only the positive solution can represent a side length.
Question 10out of 10
Expand: (x − 5)2.
Square of a difference: (x − a)² = x² − 2ax + a². The middle term is negative and the final term positive because (−a) × (−a) = a².
Time left: 00:00:00
Best results on this test
Top 10 registered students. Only each student’s best result in the period counts.
Sample questions
- Expand: (x + ___)2.
- Expand: (x − ___)2.
- Factorise: x2 − ___2.
- Factor out the common factor: ___x + ___·___.
- In (x + ___)2 = x2 + kx + ___, find coefficient k.
Algebraic identities and factorisation
The identities (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b² and a² − b² = (a − b)(a + b) help expand and factor expressions efficiently. The middle term of a squared binomial is always twice the product.
Factorisation strategy
First take out any common factor. Then look for a difference of squares or a perfect-square trinomial. Check the result by multiplying the factors back out; sign errors usually appear in the middle term.
Related mathematics tests
Linear relationships and graphs
Dynamic tasks on y = kx + b values, coefficients, points, graph properties and practical tariff models.
The Pythagorean theorem
Dynamic tasks on legs, hypotenuse, rectangle diagonals, coordinate distance and practical applications.
Systems of linear equations
Dynamic tasks on solving and checking systems, determining the number of solutions and modelling word problems.
Operations with square roots
Practise adding, subtracting and multiplying radicals: like terms, extracting square factors, signs and parentheses. Dynamic questions explain common mistakes.
The test is finished
Your test result is
????
We've marked wrong answers in red.
🏆
Congratulations! This is your personal best on this test!
Try the test again and aim to improve your score!
What's next?
Answered all questions ahead of schedule.
You can finish the test now or review your answers and complete the test later by pressing the appropriate button.