# Algebra ➤

#### Factoring Cubic Polynomials (+Questions)

Factoring polynomials is necessary for solving many types of math problems. Sometimes it becomes challenging when we encounter a cubic polynomial. One way is to find the roots by applying the cubic formula, but it is too complex to remember and use. However, ... more

#### Eigenspace (with Examples)

What is Eigenspace? Eigenspace is the span of a set of eigenvectors. These vectors correspond to one eigenvalue. So, an eigenspace always maps to a fixed eigenvalue. It is also a subspace of the original vector space. Finding it is equivalent to calculating ... more

#### Challenging Speed, Time, Distance Questions (Hints & Answers)

✩ You must already know: Speed, Time, Distance Formula : Speed=TimeDistance​ The Pythagoras Theorem Quadratic Equations Questions Try to use a minimum number of hints to reach the solution. All the best! If you find these challenging, learn to solve the ... more

# Numbers ➤

#### Examples of Irrational Numbers (With Lists)

Have you ever wondered how to create an irrational number? There are infinite irrational numbers between any two rational numbers, yet they are difficult to find. No list enumerates all the irrational numbers. There are more irrational numbers than rational ... more

#### Square root of a Prime (5) is Irrational (Proof + Questions)

This proof works for any prime number: 2, 3, 5, 7, 11, etc. Let’s prove for 5. First, we will assume that the square root of 5 is a rational number. Next, we will show that our assumption leads to a contradiction. Let us assume √5 is a rational ... more

#### Product of rational & irrational numbers is irrational (Proof + Questions)

Proof We prove this by contradiction. Let yx​ be a rational number and p an irrational number. Let us assume that product of these numbers is a rational number ba​. We are hoping to get a contradiction due to this assumption. The assumption results in the ... more

# Geometry ➤

#### Pythagoras Theorem Questions (with Answers)

Pythagoras Theorem In a right triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides. Length of the hypotenuse is c The hypotenuse is the longest side Lengths of the other sides are a, b Right Triangle Questions – ... more

#### James Garfield Pythagorean Theorem (Illustration & Proof)

Mr. James Abram Garfield proposed this proof in 1876. The surprising part is that Mr. Garfield was a politician! And not an ordinary one, he became the 20th president of the US. Wow! He proved the Pythagoras theorem with the help of geometrical construction ... more

#### Difficult Pythagoras Questions (Year 10, Guided Answers)

✩ You must already know The Pythagoras Theorem Area of Triangles Simple Algebraic Equations Questions Try to use a minimum number of hints to reach the solution. All the best! If you find these challenging, learn to solve the Pythagoras problems here. ... more