Solving Using the Quadratic Formula

Introduction

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable $x$, with the standard form: $$ ax^2 + bx + c = 0 $$ where $a$, $b$, and $c$ are coefficients, and $a \neq 0$. The solutions to a quadratic equation are the values of $x$ that satisfy the equation, commonly referred to as the roots of the equation.

The Quadratic Formula

The quadratic formula provides a direct method to find the roots of any quadratic equation. It is derived from the process of completing the square and is given by: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Here, the discriminant ($\Delta$) is the expression under the square root: $$ \Delta = b^2 - 4ac $$ The discriminant indicates the nature of the roots:

• If $\Delta > 0$, the equation has two distinct real roots.
• If $\Delta = 0$, the equation has one real root (a repeated root).
• If $\Delta < 0$, the equation has two complex conjugate roots.

Derivation of the Quadratic Formula

Deriving the quadratic formula involves completing the square on the general quadratic equation: $$ ax^2 + bx + c = 0 $$ 1. **Divide by $a$** (assuming $a \neq 0$): $$ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 $$ 2. **Move the constant term to the other side**: $$ x^2 + \frac{b}{a}x = -\frac{c}{a} $$ 3. **Complete the square** by adding $\left(\frac{b}{2a}\right)^2$ to both sides: $$ x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 $$ 4. **Simplify and solve for $x$**: $$ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} $$ $$ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} $$ $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Applications of the Quadratic Formula

The quadratic formula is versatile and applicable in various fields, including physics, engineering, economics, and biology. It is particularly useful when factoring is difficult or impossible. Examples of its application include:

• Determining projectile motion in physics.
• Calculating profit maximization in economics.
• Analyzing quadratic relationships in biological populations.

Example Problems

Example 1: Solve the quadratic equation $2x^2 - 4x - 6 = 0$ using the quadratic formula.

1. Identify coefficients: $a = 2$, $b = -4$, $c = -6$.
2. Calculate the discriminant: $$ \Delta = (-4)^2 - 4(2)(-6) = 16 + 48 = 64 $$
3. Apply the quadratic formula: $$ x = \frac{-(-4) \pm \sqrt{64}}{2(2)} = \frac{4 \pm 8}{4} $$
4. Simplify: $$ x = \frac{4 + 8}{4} = 3 \quad \text{and} \quad x = \frac{4 - 8}{4} = -1 $$

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

1. Identify coefficients: $a = 1$, $b = 2$, $c = 5$.
2. Calculate the discriminant: $$ \Delta = 2^2 - 4(1)(5) = 4 - 20 = -16 $$
3. Since $\Delta < 0$, the roots are complex: $$ x = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i $$

Common Mistakes to Avoid

• Incorrectly calculating the discriminant, leading to wrong conclusions about the nature of the roots.
• Misapplying the quadratic formula by forgetting to divide by $2a$.
• Neglecting to simplify the square root of the discriminant correctly, especially when it involves negative values.

Tips for Mastery

• Always double-check the coefficients before applying the formula.
• Practice solving equations with different types of discriminants to gain confidence.
• Understand the relationship between the discriminant and the graph of the quadratic function.

Comparison Table

Summary and Key Takeaways