Lesson 1: Getting Started With Quadratics

Property

An equation of the form $ax^2 + bx + c = 0$ is called a quadratic equation.

Examples

• The equation $x^2 - 5x + 6 = 0$ is a quadratic equation in standard form with $a=1$, $b=-5$, and $c=6$.
• The equation $4y^2 = 12$ is a quadratic equation. We can write it in standard form as $4y^2 - 0y - 12 = 0$.
• The equation $k(k-3) = 10$ is also a quadratic equation. Expanding it gives $k^2 - 3k = 10$, which becomes $k^2 - 3k - 10 = 0$ in standard form.

Explanation

A quadratic equation is a polynomial equation where the variable's highest power is two (it's squared). Unlike linear equations, they often have two solutions. The condition $a \neq 0$ is crucial, otherwise, it's not a quadratic equation.

Property

Taking a square root is the opposite of squaring a number.
To solve an equation of the form $x^2 = k$ (where $k > 0$), we take the square root of both sides.
Because a positive number has two square roots, the solution is written as:

Examples

• To solve the equation $x^2 = 81$, we take the square root of both sides. The solutions are $x = \pm\sqrt{81}$, which means $x = 9$ and $x = -9$.