Introduction
Quadratic equations and inequalities form a crucial part of the JEE Main Mathematics syllabus. They are fundamental concepts that are used in various applications in algebra, calculus, and even geometry. This study note will delve into the nuances of quadratic equations and inequalities, breaking down complex ideas into digestible sections, and providing examples to illustrate key points.
Quadratic Equations
Definition
A quadratic equation is a second-degree polynomial equation in a single variable $x$ with the form: $$ ax^2 + bx + c = 0 $$ where $a$, $b$, and $c$ are constants, and $a \neq 0$.
Standard Form
The standard form of a quadratic equation is: $$ ax^2 + bx + c = 0 $$
Roots of a Quadratic Equation
The solutions to the quadratic equation are called the roots. The roots can be found using the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
Discriminant
The discriminant $\Delta$ of the quadratic equation $ax^2 + bx + c = 0$ is given by: $$ \Delta = b^2 - 4ac $$
The discriminant helps to determine the nature of the roots:
- If $\Delta > 0$, the equation has two distinct real roots.
- If $\Delta = 0$, the equation has exactly one real root (a repeated root).
- If $\Delta
< 0$, the equation has two complex conjugate roots.
TipAlways check the discriminant first to understand the nature of the roots before solving the quadratic equation.
Methods to Solve Quadratic Equations
- Factoring:
- Express the quadratic equation in a factored form $(x - \alpha)(x - \beta) = 0$.
- Solve for $x$ by setting each factor to zero.
- Completing the Square:
- Rewrite the equation in the form $(x - h)^2 = k$ and solve for $x$.
- Quadratic Formula:
- Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Solve the quadratic equation $2x^2 - 4x - 6 = 0$ using the quadratic formula.
Here, $a = 2$, $b = -4$, and $c = -6$.
The discriminant $\Delta = (-4)^2 - 4 \cdot 2 \cdot (-6) = 16 + 48 = 64$.
Since $\Delta > 0$, there are two distinct real roots.
Using the quadratic formula: $$ x = \frac{-(-4) \pm \sqrt{64}}{2 \cdot 2} = \frac{4 \pm 8}{4} $$
So, the roots are: $$ x = \frac{12}{4} = 3 \quad \text{and} \quad x = \frac{-4}{4} = -1 $$
Nature of Roots
The sum and product of the roots $\alpha$ and $\beta$ of the quadratic equation $ax^2 + bx + c = 0$ can be given by:
- Sum of the roots: $\alpha + \beta = -\frac{b}{a}$
- Product of the roots: $\alpha \beta = \frac{c}{a}$
These relationships are useful for forming quadratic equations when the roots are known.
Graph of a Quadratic Equation
The graph of the quadratic equation $y = ax^2 + bx + c$ is a parabola.
- If $a > 0$, the parabola opens upwards.
- If $a
< 0$, the parabola opens downwards.
The vertex of the parabola is at: $$ \left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right) $$
The axis of symmetry is the vertical line $x = -\frac{b}{2a}$.
Quadratic Inequalities
Definition
A quadratic inequality is an inequality that involves a quadratic expression. It can be in one of the following forms:
- $ax^2 + bx + c >
0$
- $ax^2 + bx + c
< 0$
- $ax^2 + bx + c \geq 0$
- $ax^2 + bx + c \leq 0$
Solving Quadratic Inequalities
To solve quadratic inequalities, follow these steps:
- Find the roots of the corresponding quadratic equation $ax^2 + bx + c = 0$.
- Determine the intervals defined by the roots.
- Test the sign of the quadratic expression in each interval.
- Select the intervals that satisfy the inequality.
Solve the inequality $x^2 - 3x + 2 > 0$.
First, find the roots of the equation $x^2 - 3x + 2 = 0$.
Factoring, $(x - 1)(x - 2) = 0$, so the roots are $x = 1$ and $x = 2$.
These roots divide the number line into three intervals: $(-\infty, 1)$, $(1, 2)$, and $(2, \infty)$.
Test the sign of $x^2 - 3x + 2$ in each interval:
- For $x \in (-\infty, 1)$, choose $x = 0$: $0^2 - 3(0) + 2 = 2 > 0$
- For $x \in (1, 2)$, choose $x = 1.5$: $(1.5)^2 - 3(1.5) + 2 = -0.25
< 0$
- For $x \in (2, \infty)$, choose $x = 3$: $3^2 - 3(3) + 2 = 2 >
0$
Thus, the solution to the inequality is $(-\infty, 1) \cup (2, \infty)$.
Common MistakeDo not forget to test the sign of the quadratic expression in each interval. Simply finding the roots is not sufficient.
Conclusion
Understanding quadratic equations and inequalities is essential for solving a wide range of problems in algebra and calculus. By mastering the methods to solve these equations and inequalities, and by understanding the nature of their roots, you can tackle various problems with confidence. Remember to practice regularly and pay attention to common mistakes to improve your proficiency.
TipPractice solving quadratic equations and inequalities with different methods to find the one that works best for you.
