Algebraic equation facts for kids

An algebraic equation is a type of math problem where you try to find unknown numbers. These equations use letters, called variables, to stand for these unknown numbers. They also use regular numbers and math operations like addition, subtraction, multiplication, and division.

Think of an algebraic equation like a balance scale. Both sides of the equation must be equal. For example, in the equation `x + 5 = 10`, the letter 'x' is a variable. To make the equation true, 'x' must be 5.

What is an Algebraic Equation?

An algebraic equation is a statement that two expressions are equal. These expressions are usually polynomials. A polynomial is a math expression made of variables and coefficients, using only addition, subtraction, multiplication, and non-negative whole number exponents.

For example, `3x + 7 = 16` is an algebraic equation. Here, `3x + 7` is one expression, and `16` is another. We want to find the value of 'x' that makes both sides equal.

Variables and Coefficients

• Variables are the letters in an equation, like 'x' or 'y'. They represent numbers that can change or are unknown.
• Coefficients are the numbers multiplied by the variables. In `3x + 7 = 16`, the number `3` is the coefficient of 'x'.

Equivalent Equations

Two equations are called equivalent if they have the exact same solutions. This means that any number that solves the first equation will also solve the second, and vice versa.

For instance, `x + 2 = 5` and `x = 3` are equivalent equations. They both have `x = 3` as their only solution. You can often change an equation into an equivalent one by doing the same math operation to both sides (like adding the same number or multiplying by the same non-zero number).

Finding Solutions (Roots)

The solutions to an algebraic equation are the values for the variables that make the equation true. These solutions are also often called roots, especially when the equation is written so that one side is zero. For example, if you have `x - 3 = 0`, the root is `x = 3`.

When you solve an equation, you need to know what kind of numbers you are looking for.

• Integers: These are whole numbers (like -3, 0, 5). If you are only looking for integer solutions, the equation is called a Diophantine equation.
• Real Numbers: These include all numbers on the number line, like integers, fractions, and decimals (e.g., 1/2, 3.14, square root of 2).
• Complex Numbers: These are numbers that include the imaginary unit 'i' (where `i * i = -1`). Every algebraic equation has at least one solution if we allow complex numbers.

A Brief History of Solving Equations

People have been trying to solve algebraic equations for thousands of years!

Ancient Discoveries

• The ancient Egyptians (around 1650 BC) knew how to solve some basic equations, including those where the highest power of the variable was 2 (called "degree 2" or quadratic equations). They often used methods that were like "guess and check" or specific steps for certain problems.

Renaissance Breakthroughs

During the Renaissance in Europe, mathematicians made huge progress:

• In the 1500s, an Italian mathematician named Gerolamo Cardano found a way to solve equations where the highest power was 3 (called cubic equations).
• Soon after, Lodovico Ferrari, one of Cardano's students, figured out how to solve equations where the highest power was 4 (called quartic equations).

These solutions were often given as "radical expressions," which means they involved square roots, cube roots, and so on. For example, the positive solution to `x^2 + x - 1 = 0` is `x = (1 + sqrt(5)) / 2`.

The Limits of Radicals

For a long time, mathematicians hoped to find similar formulas using radicals for equations of degree 5 and higher.

• However, in 1824, a brilliant Norwegian mathematician named Niels Henrik Abel proved that it's not always possible to solve equations of degree 5 (called quintic equations) or higher using only radicals. This was a huge discovery!
• Later, a French mathematician named Évariste Galois developed a whole new area of math called Galois theory. This theory helps us understand exactly when an equation can be solved using radicals and when it cannot.

Algebraic equations are a fundamental part of mathematics and are used in many fields, from science and engineering to economics and computer science.

See also

In Spanish: Ecuación algebraica para niños