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