Lesson 2.2: Quadratic Equations

Introduction

In this lesson, students, we will explore the fascinating world of quadratic equations. Quadratic equations are polynomial equations of degree 2, and they take the general form:

$$

ax^2 + bx + c = 0,

$$

where $a$, $b$, and $c$ are constants with $a \neq 0$. The solutions to these equations are known as the roots, and they can be found using various methods, including factoring, completing the square, and the quadratic formula. By the end of this lesson, you will be able to confidently solve quadratic equations and determine the nature of their roots using the discriminant.

Learning Objectives

• Solve quadratic equations by factorising, completing the square, and using the quadratic formula.
• Understand the discriminant as a test for the number and nature of the roots.
• Choose an efficient method for a given quadratic equation.
• Solve a quadratic equation using the appropriate method.
• Use the discriminant to determine the number and nature of the roots.

Section 1: Understanding Quadratic Equations

Quadratic equations are essential in mathematics and appear in various applications, from physics to economics. To understand these equations, let's break them down further.

Standard Form

As mentioned earlier, the standard form of a quadratic equation is:

$$

ax^2 + bx + c = 0.

$$

Terms Explained:

• $a$: The coefficient of $x^2$, determining the shape of the parabola. It cannot be zero.
• $b$: The coefficient of $x$, affecting the position of the vertex.
• $c$: The constant term, which shifts the equation up or down.

Graphical Representation

The graph of a quadratic equation is a parabola, which opens upwards if $a > 0$ and downwards if $a < 0$. The point where the parabola crosses the x-axis represents the roots of the equation.

Example: For the quadratic equation $2x^2 - 4x - 6 = 0$, the parabola opens upwards since the coefficient of $x^2$ is positive.

Finding the Roots

Quadratic equations can have two distinct real roots, one repeated real root, or no real roots at all. The nature of the roots can be determined using the discriminant:

$$\Delta = b^2 - 4ac.

$$

• Two distinct real roots if $\Delta > 0$.
• One repeated real root if $\Delta = 0$.
• No real roots if $\Delta < 0$.

Section 2: Solving Quadratic Equations by Factorising

One of the simplest methods for finding the roots of a quadratic equation is by factorising it. This method is effective when the quadratic can be expressed as a product of binomials.

Steps for Factorisation

1. Write the equation in standard form: $ax^2 + bx + c = 0$.
2. Factorise the quadratic into the form $(px + q)(rx + s) = 0$.
3. Set each factor equal to zero and solve for $x$.

Example: Solve the equation $x^2 - 5x + 6 = 0$ by factorising.

1. The equation is already in standard form.
2. Factorise: $(x - 2)(x - 3) = 0$.
3. Set each factor to zero:

• $x - 2 = 0 \Rightarrow x = 2$.
• $x - 3 = 0 \Rightarrow x = 3$.

Thus, the roots are $x = 2$ and $x = 3$.

Section 3: Solving Quadratic Equations by Completing the Square

Completing the square is another method that can be used to solve quadratic equations, and it also gives insight into the vertex of the parabola.

Steps for Completing the Square

1. Start with the equation $ax^2 + bx + c = 0$.
2. Divide through by $a$ (if $a \neq 1$).
3. Rearrange the equation to isolate the constant term on one side:

$\ n = -\frac{b}{2a} $

1. Add and subtract $(\frac{b}{2a})^2$ to complete the square.
2. Solve for $x$.

Example: Solve the equation $x^2 - 4x + 3 = 0$ by completing the square.

1. The equation is in standard form.
2. Rearranging gives us:

$$x^2 - 4x = -3.$$

1. Add and subtract $(\frac{4}{2})^2 = 4$:

$$x^2 - 4x + 4 = 1.$$

1. This can be rewritten as:

$$(x - 2)^2 = 1.$$

1. Taking square roots:

$$x - 2 = 1 \quad \text{or} \quad x - 2 = -1$$

Thus, $x = 3$ or $x = 1$.

Section 4: Solving Quadratic Equations Using the Quadratic Formula

The quadratic formula provides a straightforward way to find the roots of any quadratic equation. It is particularly useful when the equation does not factor easily.

The Quadratic Formula

The quadratic formula is given by:

$$

x = $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

$$

Steps to Use the Quadratic Formula

1. Identify the values of $a$, $b$, and $c$ from the equation $ax^2 + bx + c = 0$.
2. Calculate the discriminant $\Delta = b^2 - 4ac$.
3. Substitute $a$, $b$, and $\Delta$ into the quadratic formula to find the roots.

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

1. Identify $a = 1$, $b = -3$, and $c = 2$.
2. Calculate the discriminant:

$$\Delta = (-3)^2 - 4(1)(2) = 9 - 8 = 1.$$

1. Substitute into the formula:

$$x = \frac{-(-3) \pm \sqrt{1}}{2(1)} = \frac{3 \pm 1}{2}.$$

Thus, the two roots are:

• $x = \frac{4}{2} = 2$.
• $x = \frac{2}{2} = 1$.

Section 5: The Discriminant and Its Significance

The discriminant not only allows us to find the roots of a quadratic equation, but it also helps us understand the nature and number of roots.

Analyzing the Discriminant

i. If $\Delta > 0$, there are two distinct real roots.

ii. If $\Delta = 0$, there is one repeated real root.

iii. If $\Delta < 0$, there are no real roots (the roots are complex).

Example of Discriminant Analysis

Consider the equation $2x^2 + 4x + 2 = 0$.

1. Identify $a = 2$, $b = 4$, and $c = 2$.
2. Calculate the discriminant:

$$\Delta = 4^2 - 4(2)(2) = 16 - 16 = 0.$$

1. Since $\Delta = 0$, there is exactly one repeated real root.

This means the parabola touches the x-axis at one point, which is the vertex.

Conclusion

In this lesson, students, we have explored the various methods of solving quadratic equations, including factorising, completing the square, and using the quadratic formula. We have also discussed the discriminant and its role in determining the nature and number of roots. Each method has its advantages and is suitable for different scenarios. Mastering these techniques will enhance your ability to work with not just quadratic equations, but broader mathematical problems as well.

Study Notes

• A quadratic equation is in standard form: $ax^2 + bx + c = 0$.
• Roots of the equation can be found using:
• Factorisation.
• Completing the square.
• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• The discriminant $\Delta = b^2 - 4ac$ indicates:
• Two distinct real roots if $\Delta > 0$.
• One repeated real root if $\Delta = 0$.
• No real roots if $\Delta < 0$.
• The choice of method depends on the specific quadratic equation and personal preference.