Math Academy turns difficult math topics into clear lessons, step-by-step explanations, and practical exercises so students can build real understanding instead of memorizing formulas.
Math Academy murakkab matematik mavzularni tushunarli darslar, bosqichma-bosqich izohlar va amaliy mashqlar orqali o‘rgatadi. Maqsad — formulalarni yodlash emas, haqiqiy tushuncha hosil qilish.
Lesson progress
Darsdagi progress
Understand concepts with visual examples and guided practice.
Tushunchalarni vizual misollar va mashqlar orqali o‘rganing.
Learn first, practice after understanding.
Avval tushuning, keyin mashq qiling.
The platform combines structured lessons, visual explanations, and active practice in one place.
Platforma tartibli darslar, vizual tushuntirishlar va faol mashqlarni bitta joyda birlashtiradi.
Topics are organized like a guided textbook, helping students move from basic ideas to harder concepts step by step.
Mavzular darslikka o‘xshash tartibda tuzilgan: o‘quvchi oddiy g‘oyalardan murakkab tushunchalarga bosqichma-bosqich o‘tadi.
Instead of only giving formulas, lessons explain why the method works and how students should think through each problem.
Darslar faqat formulani bermaydi; metod nima uchun ishlashini va masalaga qanday fikrlash kerakligini tushuntiradi.
English and Uzbek support helps students learn math while becoming familiar with academic English terminology.
Inglizcha va o‘zbekcha qo‘llab-quvvatlash o‘quvchilarga matematika bilan birga akademik inglizcha terminlarni ham o‘rganishga yordam beradi.
Open these lessons directly on this page before creating an account.
Ro‘yxatdan o‘tishdan oldin ushbu darslarni shu sahifaning o‘zida ochib ko‘ring.
Create an account to access full lessons, practice questions, and your personal learning progress.
To‘liq darslar, amaliy savollar va shaxsiy progressingizga kirish uchun akkaunt yarating.
A fraction is a number that describes how many equal parts of a whole are being considered. It is written with a numerator on top and a denominator on the bottom.
The whole is divided into 2 equal parts, and 1 part is taken.
The whole is divided into 4 equal parts, and 3 parts are taken.
A fraction is not only a “piece of a pizza.” In mathematics, the same fraction can represent several ideas.
| Meaning | Example | Explanation |
|---|---|---|
| Part of a whole | \(\dfrac{3}{5}\) | 3 of 5 equal parts are selected. |
| Division | \(\dfrac{3}{5}=3\div5\) | The numerator is divided by the denominator. |
| Ratio | \(\dfrac{3}{5}\) | 3 parts are compared with 5 parts. |
| Point on a number line | \(\dfrac{3}{5}=0.6\) | The value is located between 0 and 1. |
Before solving fraction problems, identify the type of fraction. This helps you choose the right method.
The numerator is smaller than the denominator. Its value is between 0 and 1.
The numerator is greater than or equal to the denominator. Its value is at least 1.
A whole number and a proper fraction written together.
To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the same denominator.
To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient becomes the whole-number part. The remainder becomes the numerator of the fractional part. The denominator stays the same.
Compute the division \(17\div4\).
Find the greatest whole number of groups of \(4\) inside \(17\). Since \(4\cdot4=16\) and \(4\cdot5=20\), the quotient is \(4\).
Find the remainder: \(17-16=1\). This leftover \(1\) becomes the numerator of the fractional part.
Keep the original denominator \(4\). Therefore, the mixed number is \(4\dfrac{1}{4}\).
Equivalent fractions have the same value even though they look different. You create equivalent fractions by multiplying or dividing the numerator and denominator by the same nonzero number.
\(\dfrac{1}{2}\)
\(\dfrac{2}{4}\)
A fraction is in simplest form when the numerator and denominator have no common factor except 1. The fastest method is to divide both by their greatest common divisor.
Find the greatest common divisor of 12 and 18.
You can also divide by smaller common factors until nothing remains.
The negative sign can be written in front of the fraction, in the numerator, or in the denominator. These mean the same thing.
To compare fractions, first check whether the denominators or numerators are the same. If not, use a common denominator or cross multiplication.
When denominators are the same, the larger numerator gives the larger fraction.
When numerators are the same, the smaller denominator gives the larger fraction because each piece is bigger.
Cross multiply and compare the products.
Fractions can be added or subtracted only when they refer to equal-sized parts. That is why we need a common denominator.
Keep the denominator and add the numerators.
Find the least common denominator, rewrite both fractions, then add the numerators.
Add whole parts with whole parts and fraction parts with fraction parts. If the fraction part becomes improper, convert it.
Keep the denominator and subtract the numerators.
If the fraction part on top is too small, borrow 1 whole and turn it into a fraction with the same denominator.
Multiply numerator by numerator and denominator by denominator. Simplify before or after multiplying.
Before multiplying, you may cancel common factors across numerator and denominator.
To divide by a fraction, multiply by its reciprocal. This is often called “keep, change, flip.”
Enter any fraction. The graph shows its decimal value on a number line and displays a visual bar model. Improper fractions are allowed, so you can see values greater than 1.
Use this tool to compare two fractions. The page simplifies both fractions and compares their actual values.
To find a fraction of a number, multiply the number by the fraction.
If a fraction is used, subtract it from 1 to find the remaining part.
If a fraction of the total is known, divide the known amount by the fraction.
Do not add denominators when adding fractions.
You cannot add or subtract fractions with different denominators directly.
When dividing by a fraction, multiply by the reciprocal of the second fraction.
A linear equation has the variable mostly in the first power only.
General form:
where \(a \ne 0\).
Examples:
The graph of a linear equation is a straight line.
Whatever you do to one side, you must do to the other side.
You can:
When a term moves across the equal sign, its sign changes. A positive term becomes negative, and a negative term becomes positive.
Move \(+7\) to the right side. It becomes \(-7\).
Another quick example:
Solve:
Step 1: Subtract 5 from both sides.
Step 2: Divide both sides by 2.
Answer:
Check:
Correct.
Solve:
Step 1: Move \(x\) to the left side.
Step 2: Add 4 to both sides.
Step 3: Divide by 2.
Answer:
Check:
Left side:
Right side:
Correct.
Solve:
Step 1: Subtract 5.
Step 2: Multiply both sides by 3.
Answer:
Check:
Correct.
A linear equation with one variable has the form:
where a, b, c are constants and x is the unknown variable.
When a term crosses the equals sign, its operation flips:
Multi-step: (2x + 5) / 3 = 7
A quadratic equation has \(x^2\) as the highest power.
General form:
where \(a \ne 0\).
Examples:
The graph of a quadratic equation is a parabola.
The solutions of a quadratic equation are the \(x\)-values where the parabola crosses the \(x\)-axis.
There are three common methods:
Solve:
We need two numbers that:
Those numbers are \(-2\) and \(-3\).
So:
Now solve:
If two factors multiply to zero, at least one of them must be zero.
So:
or
Answer:
Check with \(x = 2\):
Check with \(x = 3\):
Correct.
Solve:
Take the square root of both sides.
Important: when solving \(x^2 = a\), we take both positive and negative roots.
Answer:
Why both?
Because:
and
Solve:
Here:
Use the quadratic formula:
Substitute:
Simplify inside the square root:
Now split into two answers.
First:
Second:
Answer:
A quadratic equation has the form:
Discriminant: D = b² − 4ac
Method 1 — Factoring
Find two numbers: multiply to c, add to b (when a = 1).
Method 2 — Quadratic Formula
A linear inequality is like a linear equation, but with inequality signs.
General forms:
Example:
When you multiply or divide both sides by a negative number, you must flip the inequality sign.
For example:
Divide by \(-2\), and flip \(\lt\) to \(\gt\):
This is one of the most important inequality rules.
Change the sign and values. The blue shadow shows all \(x\)-values that make the inequality true.
Solve:
Step 1: Subtract 5.
Step 2: Divide by 2.
Answer:
This means every number less than 6 works.
Examples that work:
Check with \(x = 5\):
Correct.
Solve:
Step 1: Subtract 4.
Step 2: Divide by \(-3\).
Because we divide by a negative number, flip the sign:
Answer:
Check with \(x = -5\):
Correct.
Check with \(x = 0\):
False.
So \(x \le -4\) makes sense.
Solve:
Step 1: Move \(2x\) to the left.
Step 2: Add 7.
Step 3: Divide by 3.
Answer:
Check with \(x = 4\):
Left side:
Right side:
Correct.
Same as linear equations but uses >, <, ≥, ≤ instead of =.
A quadratic inequality has \(x^2\) and an inequality sign.
Examples:
A quadratic inequality asks:
The graph is a parabola.
Change the sign and values. The blue shadow shows the solution intervals.
The main strategy:
Suppose:
The roots are:
These divide the number line into three intervals:
Then we test each interval.
Solve:
Step 1: Factor.
So:
Step 2: Find roots.
Step 3: Split number line.
Step 4: Test each interval.
Choose \(x = 0\).
Positive. So this interval works.
Choose \(x = 2.5\).
Negative. Does not work.
Choose \(x = 4\).
Positive. Works.
Answer:
Interval notation:
Important: because the inequality is \(\gt 0\), not \(\ge 0\), we do not include 2 and 3.
Solve:
Step 1: Factor.
So:
Step 2: Find roots.
Step 3: Split number line.
Step 4: Test each interval.
Choose \(x = -3\).
Positive. We need \(\le 0\), so it does not work.
Choose \(x = 0\).
Negative. This works.
Choose \(x = 3\).
Positive. Does not work.
Step 5: Include endpoints?
The inequality is:
So yes, include the roots \(-2\) and \(2\), because at those points the expression equals zero.
Answer:
Interval notation:
Solve:
Step 1: Factor.
So:
Step 2: Find roots.
and:
Step 3: Split number line.
Step 4: Test intervals.
Choose \(x = -3\).
Positive. Works.
Choose \(x = 0\).
Negative. Does not work.
Choose \(x = 1\).
Positive. Works.
Step 5: Include endpoints?
The inequality is:
So yes, include the roots.
Answer:
Interval notation:
Form: ax² + bx + c > 0 (or <, ≥, ≤)
Case 1 — x² − 5x + 6 > 0
| Interval | Test point | Sign |
|---|---|---|
| x < 2 | x = 1 | + positive ✓ |
| 2 < x < 3 | x = 2.5 | − negative ✗ |
| x > 3 | x = 4 | + positive ✓ |
Case 2 — x² − 4x + 3 ≤ 0
| Interval | Test point | Sign |
|---|---|---|
| x < 1 | x = 0 | + positive |
| 1 < x < 3 | x = 2 | − negative ✓ |
| x > 3 | x = 4 | + positive |
Example:
Answer:
An equation usually gives exact values.
Example:
Answer:
An inequality usually gives ranges of values.
| Type | Form | Number of solutions |
|---|---|---|
| Linear equation | \(ax + b = 0\) | usually one solution |
| Quadratic equation | \(ax^2 + bx + c = 0\) | usually two, one, or no real solutions |
| Linear inequality | \(ax + b \gt 0\) | usually one interval |
| Quadratic inequality | \(ax^2 + bx + c \gt 0\) | usually one or two intervals |
Example:
Add 9:
Divide by 4:
Fast.
Example:
Ask:
Answer:
So:
Example:
Roots:
A product of two factors is negative when the signs are different.
That happens between the roots:
Solution:
Solution:
Solution:
Solution:
The main mental model:
All four topics ask the same first question: what values of x make the statement true?
Module 1 — Introduction to Planimetry
Triangles can be grouped by their sides and by their angles. This lesson focuses on three very important families: right triangles, equilateral triangles, and isosceles triangles.
A right triangle is a triangle in which one angle is exactly \(90^\circ\). The other two angles are acute, so each is less than \(90^\circ\).
The two acute angles are complementary.
If the legs are \(a\) and \(b\), and the hypotenuse is \(c\), then:
Example: if \(a=6\) and \(b=8\), then \(c=\sqrt{6^2+8^2}=10\).
The area formula uses the legs because they are perpendicular.
For an acute angle \(\theta\), the basic ratios are:
An equilateral triangle is a triangle in which all three sides are equal in length. Because the sides are equal, all three interior angles are also equal.
All angles are equal and acute.
The altitude of an equilateral triangle splits it into two congruent right triangles and divides the base into two equal parts.
If \(a=10\), then \(h=5\sqrt3\).
An isosceles triangle is a triangle in which at least two sides are equal in length. The angles opposite the equal sides are also equal.
The angles opposite the equal sides are equal. These are called the base angles.
The altitude from the vertex to the base is special: it is perpendicular to the base, bisects the base, and bisects the vertex angle.
This creates two congruent right triangles.
If the height is unknown, use the Pythagorean Theorem on half of the base.
If the equal sides are \(13\) and the base is \(10\), half of the base is \(5\). The height is:
