Introduction

In mathematics, a quadratic equation is a polynomial equation of the form $ax^2 + bx + c = 0$, where $a$, $b$ and $c$ are constants and $a \neq 0$. Quadratic equations can be solved using various methods such as factoring, completing the square, or using the quadratic formula.

Definition and Example: Standard Form

A quadratic equation in standard form is of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$.

Example: Solve the quadratic equation $2x^2 - 5x + 2 = 0$.

[ \begin{aligned} 2x^2 - 5x + 2 & = 0 \ (2x - 1)(x - 2) & = 0 \quad \text{(factoring)} \ \Rightarrow x & = \frac{1}{2}, 2 \end{aligned} ]

Definition and Example: Discriminant

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by $b^2 - 4ac$. It helps determine the nature of the roots of the quadratic equation.

Example: Find the discriminant for the equation $3x^2 + 4x + 1 = 0$.

[ \begin{aligned} a & = 3, \quad b = 4, \quad c = 1 \ \text{Discriminant} & = b^2 - 4ac \ & = 4^2 - 4(3)(1) \ & = 16 - 12 \ & = 4 \end{aligned} ]

Definition and Example: Quadratic Formula

The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are coefficients of the quadratic equation $ax^2 + bx + c = 0$.

Example: Solve the quadratic equation $x^2 - 3x + 2 = 0$ using the quadratic formula.

[ \begin{aligned} a & = 1, \quad b = -3, \quad c = 2 \ x & = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(2)}}{2(1)} \ & = \frac{3 \pm \sqrt{1}}{2} \ & = \frac{3 \pm 1}{2} \ \Rightarrow x & = 2, 1 \end{aligned} ]

Common Mistakes

• Forgetting to check if $a \neq 0$ in the quadratic equation.
• Incorrectly calculating the discriminant.
• Making errors while applying the quadratic formula.

Key Points

• A quadratic equation is of the form $ax^2 + bx + c = 0$.
• The discriminant helps in determining the nature of the roots.
• The quadratic formula is used to solve quadratic equations.

Practice Questions

1. Solve the quadratic equation $4x^2 + 12x + 9 = 0$.
2. Find the discriminant for the equation $2x^2 - 5x + 3 = 0$.
3. Solve the quadratic equation $x^2 + 6x + 9 = 0$ using the quadratic formula.