# What are equations and their roots?

Haknem School
A comprehensive guide to understanding the concept of equations and their roots. Explore how to solve them and why these concepts are crucial in mathematics.

## Equation and its roots

An equation is a mathematical expression that contains an unknown quantity that we are trying to find.

For example, consider the equation x – 3 = 7. Here, x is the unknown quantity.

The root of the equation is a number that, when substituted for the unknown, turns the equation into a correct equality.

Suppose, if x = 2, then substituting this value into the equation, we get 2 – 3 = 7. It turns out, -1 = 7, which is an incorrect equality. Therefore, x = 2 is not a root of the equation.

But if x = 10, then substituting this value into the equation, we get 10 – 3 = 7. It turns out, 7 = 7, which is a correct equality. Therefore, x = 10 is a root of the equation.

To solve an equation means to find all its roots or to prove that they do not exist.

A quadratic equation is an equation of the form ax² + bx + c = 0, where a is the coefficient at x² (it should not be zero), b is the coefficient at x, c is the free term.

To better remember how the coefficients are located, let’s try to determine them on examples.

• Equation: 2x² – 5x + 3 = 0
Coefficients:
a = 2
b = -5
c = 3
• Equation: -3x² + 4x – 2 = 0
Coefficients:
a = -3
b = 4
c = -2
• Equation: 6x² + 7x – 1 = 0
Coefficients:
a = 6
b = 7
c = -1

Now that you know how to determine the coefficients in a quadratic equation, you are ready for more complex tasks!

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