Quadratic Equation

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in the form: \(ax^2 + bx + c = 0\)

where:

- \( a, b, c \) are constants, with \( a \neq 0 \),

- \( x \) represents the variable.

It has only one variable with degree 2. The quadratic equation has a single variable.

Solution of Quadratic Equation

The solution of a quadratic equation \(ax^2 + bx + c = 0\) refers to the values of \(x\) that satisfy the equation. These values are also called roots or zeroes of the equation. It has two roots, meaning that the quadratic equation has two values of \(x\) that satisfy the equation. The quadratic equation has two roots.

Example:

Q. Find the roots of the quadratic equation \(x^2 + 5x + 6 = 0\).

Hence, the roots of the equation \(x^2 + 5x + 6 = 0\) are \(x = -3\) and \(x = -2\). If you put these values in the original equation \(x^2 + 5x + 6 = 0\) the result will be \(0 = 0\). Hence, the solution satisfies the equation.

Methods to Solve a Quadratic Equation

There are several methods to solve quadratic equations. In this class, we will learn methods like factoring, completing the square, and using the quadratic formula.

1. Factoring 

   - Express the quadratic equation in factored form: 

     \((x - p)(x - q) = 0\)

   - Solve for \( x \) by setting each factor to zero: 

     \(x - p = 0 \quad \text{or} \quad x - q = 0\)

2. Completing the Square 

   - Rewrite the equation in the form: 

     \((x - h)^2 = k\)

   - Solve for \( x \) by taking the square root.

3. Quadratic Formula 

   - Use the formula: 

     \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

   - This method works for all quadratic equations, even if they are not factorable.