Algebra and its notations look intimidating mostly because the symbols are unfamiliar, not because the ideas are hard. This guide explains the quadratic formula, scientific notation, matrices and binary — what each is for, and how to read it.
A quadratic equation has the form ax² + bx + c = 0, where the highest power is 2. The quadratic formula solves any of them.
Take x² − 5x + 6 = 0, so a = 1, b = −5, c = 6. Substituting: the part under the square root is (−5)² − 4(1)(6) = 25 − 24 = 1. So x = (5 ± 1) ÷ 2, giving x = 3 or x = 2. You can check by substituting both back in — each makes the equation zero.
The ± symbol is why a quadratic usually has two solutions: one using plus, one using minus. Our quadratic formula calculator shows the substitution and both roots.
The part under the square root, b² − 4ac, is called the discriminant, and its sign tells you how many real solutions exist before you finish the calculation.
Scientific notation writes very large or very small numbers compactly, as a number between 1 and 10 multiplied by a power of ten. It is how science, engineering and calculators handle numbers that would otherwise be a wall of zeros.
93,000,000 becomes 9.3 × 10⁷ — the exponent 7 counts how many places the decimal point moved left. A small number moves the other way: 0.00042 becomes 4.2 × 10⁻⁴, where the negative exponent counts places moved right.
The rule of thumb: a positive exponent means a large number, a negative exponent means a small one, and the exponent is simply how many places the decimal point shifted. Our scientific notation calculator converts both ways.
A matrix is a rectangular grid of numbers, and it exists to handle many equations or many data points at once. Rather than solving a system of equations one variable at a time, a matrix lets you operate on the whole system together.
This is why matrices sit behind so much modern computing: 3D graphics, the transformations that rotate and scale an image, and the linear algebra underneath machine learning are all matrix operations. Adding matrices is done element by element; multiplying them follows a specific row-by-column rule that our matrix calculator carries out step by step.
The key idea to carry away is that a matrix is a container for structured numbers, and matrix operations are shortcuts for doing the same thing to all of them consistently.
We count in base 10 — ten digits, 0 to 9 — probably because we have ten fingers. Computers count in base 2, using only 0 and 1, because a circuit is either off or on.
In binary, each position is a power of two rather than a power of ten. The binary number 1011 is (1×8) + (0×4) + (1×2) + (1×1) = 11 in our usual base 10. Hexadecimal (base 16) is a compact shorthand programmers use because it maps neatly onto binary — one hex digit equals exactly four binary digits.
None of this changes the numbers themselves; it only changes how they are written. Our binary calculator and hexadecimal calculator convert between bases and show the working.
Before any of these notations, one rule governs every calculation: the order in which operations are carried out. Get it wrong and a correct formula still gives a wrong answer.
The convention is often remembered as BODMAS or BIDMAS: Brackets, Orders (powers and roots), Division and Multiplication, then Addition and Subtraction. Division and multiplication rank equally and are worked left to right, as do addition and subtraction.
Multiplication is done before addition, so 3 × 4 is evaluated first. To force the addition first, you need brackets: (2 + 3) × 4 = 20. This is why brackets exist — they override the default order.
How do I solve a quadratic equation?
Put it in the form ax² + bx + c = 0, then apply the quadratic formula: x equals negative b, plus or minus the square root of (b² − 4ac), all divided by 2a. For x² − 5x + 6 = 0 this gives x = 3 or x = 2. The plus-or-minus is why there are usually two solutions.
What does the discriminant tell me?
The discriminant is b² − 4ac, the part under the square root. If it is positive there are two real solutions, if zero there is one repeated solution, and if negative there are no real solutions because you cannot take the square root of a negative number in the real numbers.
How do I write a number in scientific notation?
Write it as a number between 1 and 10 multiplied by a power of ten. Move the decimal point until one non-zero digit remains before it, and the number of places moved becomes the exponent. 93,000,000 is 9.3 × 10⁷; 0.00042 is 4.2 × 10⁻⁴.
What is a matrix used for?
A matrix is a grid of numbers that lets you handle many equations or data points at once. Matrices power 3D graphics, image transformations, and the linear algebra behind machine learning, because they apply the same operation consistently across structured sets of numbers.
Why do computers use binary?
Because a circuit has two stable states, off and on, which map naturally to 0 and 1. Binary is base 2, where each position is a power of two. The binary number 1011 equals 11 in base 10. Hexadecimal is a compact shorthand where one digit represents four binary digits.
Does converting between number systems change the value?
No. Binary, decimal and hexadecimal are just different ways of writing the same number. The value 11 in base 10 is 1011 in binary and B in hexadecimal — the same quantity, written with different symbols and place values.