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Quadratic equations and their solutions

A quadratic equation (or second-degree equation) is a polynomial equation in which the highest exponent of the variable is 2.

General form:

It is written as:

$ ax^2 + bx + c = 0 $

Where:

a, b, c → are constant numbers (with $ a \neq 0 $).

x → is the unknown variable.

How to find the solutions?

First, we calculate the discriminant (Δ):

$ \Delta = b^2 - 4ac $

It determines how many solutions the equation has.

Types of solutions:

1. Two real and different roots:

When: $ \Delta > 0 $

✔️ The equation has two distinct real solutions.

✔️ The x values are different.

2. Two equal real roots:

When: $ \Delta = 0 $

✔️ The equation has one repeated solution.

✔️ Both roots are equal.

3. No real roots:

When: $ \Delta < 0 $

❌ There are no real solutions.

✔️ The solutions are complex numbers.

Simple summary:

Second-degree equation: $ ax^2 + bx + c = 0 $.

Discriminant: $ \Delta = b^2 - 4ac $.

Types of solutions:

$ \Delta > 0 $ → 2 different roots.

$ \Delta = 0 $ → 1 double root.

$ \Delta < 0 $ → no real roots.

Did you know?

A matemática para mim é uma matéria fascinante. Adoro desafios e na época de escola essa matéria da equação de segundo grau tive várias dificuldades, mais depois que aprendi, percebi que ela tem diversas aplicações em várias áreas da matemática e da ciência, sendo fundamental para resolver problemas e modelar fenômenos do mundo real. Vale apena aprender. by ＠LuciaMara, Brasil

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