Solution in radicals facts for kids

A solution in radicals is a way to find the answer to a math problem, specifically a polynomial equation. It means you can write the answer using only basic math operations like addition, subtraction, multiplication, and division. You can also use powers (like x²) and roots (like square roots or cube roots). Think of it as finding the answer using only the tools on a basic calculator, plus a root button!

A famous example is how you solve a quadratic equation. This is an equation that looks like ax² + bx + c = 0. The solution, which uses radicals, is:

x = (-b ± √(b² - 4ac)) / 2a

You might have seen this formula before! It uses addition, subtraction, multiplication, division, and a square root (which is a type of radical).

What Are Radicals?

Radicals are simply the mathematical term for roots.

• A square root (like √9 = 3) is a radical.
• A cube root (like ³√8 = 2) is also a radical.
• Any nth root (like ⁴√16 = 2) is a radical.

So, a "solution in radicals" means the answer can be written using these kinds of roots and basic math.

Equations That Can Be Solved With Radicals

You already know that quadratic equations (equations with x²) can be solved using radicals. Math experts have also found ways to solve cubic equations (equations with x³) and quartic equations (equations with x⁴) using similar, but more complicated, radical formulas.

For example, if you have an equation like x¹⁰ = 2, you can easily find a solution using a radical: x = ¹⁰√2. This is a solution in radicals!

When Radicals Don't Work

For a long time, mathematicians tried to find similar formulas for equations with even higher powers, like quintic equations (equations with x⁵). However, it turns out that for most equations with x⁵ or higher, you cannot find a solution using only radicals.

A famous mathematician named Évariste Galois (who lived a very short but brilliant life!) proved this. His work, called Galois theory, showed exactly which equations can be solved with radicals and which cannot. For example, an equation like x⁵ - x + 1 = 0 cannot be solved using only radicals.

Even though some equations can't be solved with radicals, they still have answers! These answers just can't be written down using only the basic operations and roots. Sometimes, you need more advanced math tools, like transcendental functions (which are special functions like logarithms or trigonometric functions).

See also

In Spanish: Solución algebraica para niños

• Solvable quintics
• Solvable sextics
• Solvable septics