# Nakafa Framework: LLM

URL: https://nakafa.com/en/subject/high-school/10/mathematics/quadratic-function/quadratic-equation
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/quadratic-function/quadratic-equation/en.mdx

Output docs content for large language models.

---

import { ReadingRoomProblem } from "./reading-room-problem";

export const metadata = {
  title: "Quadratic Equations",
  description: "Learn to solve quadratic equations using factorization, completing the square, and quadratic formula. Master ax²+bx+c=0 with step-by-step examples and real-world problems.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/19/2025",
  subject: "Quadratic Functions",
};

## What is a Quadratic Equation?

A quadratic equation is a mathematical equation involving a quadratic form. This equation contains a variable with the highest power of 2. The general form of a quadratic equation is:

<BlockMath math="ax^2 + bx + c = 0" />

with the condition that <InlineMath math="a \neq 0" /> and <InlineMath math="a, b, c" /> are real numbers.

### Origins of the Term "Quadratic"

The term "quadratic" comes from the Latin word _quadratus_, which means "to make a square." This relates to the geometric interpretation of the form <InlineMath math="x^2" /> which can be viewed as the area of a square with side length <InlineMath math="x" />.

## How to Solve Quadratic Equations

Quadratic equations can be solved in various ways. Here are some commonly used methods:

### Factorization

The factorization method involves breaking down the quadratic equation into a product of two linear factors. For example:

<div className="flex flex-col gap-4">
  <BlockMath math="2x^2 - 3x - 2 = 0" />
  <BlockMath math="(2x + 1)(x - 2) = 0" />
</div>

From the factored form above, we can get the solutions:

- If <InlineMath math="2x + 1 = 0" />, then <InlineMath math="x = -\frac{1}{2}" />
- If <InlineMath math="x - 2 = 0" />, then <InlineMath math="x = 2" />

Therefore, the roots of the quadratic equation are <InlineMath math="x = -\frac{1}{2}" /> or <InlineMath math="x = 2" />.

### Completing the Square

This method involves transforming the quadratic equation into a perfect square form.

Example:

<BlockMath math="2x^2 - 3x - 2 = 0" />

We divide all terms by 2:

<BlockMath math="x^2 - \frac{3}{2}x - 1 = 0" />

Move the constant to the right side:

<BlockMath math="x^2 - \frac{3}{2}x = 1" />

Add <InlineMath math="\left(\frac{-3/2}{2}\right)^2 = \frac{9}{16}" /> to both sides:

<div className="flex flex-col gap-4">
  <BlockMath math="x^2 - \frac{3}{2}x + \frac{9}{16} = 1 + \frac{9}{16}" />
  <BlockMath math="\left(x - \frac{3}{4}\right)^2 = \frac{16 + 9}{16} = \frac{25}{16}" />
  <BlockMath math="\left(x - \frac{3}{4}\right)^2 = \left(\frac{5}{4}\right)^2" />
</div>

Therefore:

<div className="flex flex-col gap-4">
  <BlockMath math="x - \frac{3}{4} = \pm \frac{5}{4}" />
  <BlockMath math="x = \frac{3}{4} \pm \frac{5}{4}" />
  <BlockMath math="x = 2 \text{ or } x = -\frac{1}{2}" />
</div>

### Using the Quadratic Formula

For the equation <InlineMath math="ax^2 + bx + c = 0" />, the roots can be determined using the formula:

<BlockMath math="x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}" />

Example:

<BlockMath math="2x^2 - 3x - 2 = 0" />

With <InlineMath math="a = 2" />, <InlineMath math="b = -3" />, and <InlineMath math="c = -2" />:

<div className="flex flex-col gap-4">
  <BlockMath math="x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-2)}}{2(2)}" />
  <BlockMath math="x = \frac{3 \pm \sqrt{9 + 16}}{4}" />
  <BlockMath math="x = \frac{3 \pm \sqrt{25}}{4}" />
  <BlockMath math="x = \frac{3 \pm 5}{4}" />
</div>

Therefore:

<BlockMath math="x = \frac{3 + 5}{4} = 2 \text{ or } x = \frac{3 - 5}{4} = -\frac{1}{2}" />

## Formulating Problems as Quadratic Equations

Many real-life problems can be modeled using quadratic equations. Let's explore some examples:

### Reading Room Problem

Four reading corners of the same size are created in a classroom measuring 4 m × 6 m. If each corner is a square with side length <InlineMath math="x" /> meters, then the remaining area of the room for arranging student seating is:

<div className="flex flex-col gap-4">
  <BlockMath math="\text{Total area} - \text{Area of four corners}" />
  <BlockMath math="4 \times 6 - 4x^2 = 24 - 4x^2" />
</div>

<ReadingRoomProblem />

### Problem of Multiplying Two Numbers

The product of two numbers is 63 and their sum is 16. We can solve this using a quadratic equation.

Let's say the two numbers are <InlineMath math="p" /> and <InlineMath math="q" />, then:

- <InlineMath math="p + q = 16" />, so <InlineMath math="q = 16 - p" />
- <InlineMath math="p \times q = 63" />

Substituting the value of <InlineMath math="q" />:

<div className="flex flex-col gap-4">
  <BlockMath math="p(16 - p) = 63" />
  <BlockMath math="16p - p^2 = 63" />
  <BlockMath math="-p^2 + 16p - 63 = 0" />
  <BlockMath math="p^2 - 16p + 63 = 0" />
</div>

By factoring this equation or using the quadratic formula, we can find the values of <InlineMath math="p" /> and <InlineMath math="q" />.

### Vehicle Speed Problem

A vehicle travels a distance of 320 km at a certain speed. If the vehicle travels 24 km/h faster, its travel time is reduced by 3 hours. We can find the initial speed using a quadratic equation.

Let's say the initial speed is <InlineMath math="v" /> km/h and the initial travel time is <InlineMath math="t" /> hours, then:

- <InlineMath math="v \times t = 320" /> (distance = speed × time)
- <InlineMath math="(v + 24) \times (t - 3) = 320" /> (second condition)

From the first equation: <InlineMath math="t = \frac{320}{v}" />

Substituting into the second equation:

<BlockMath math="(v + 24) \times \left(\frac{320}{v} - 3\right) = 320" />

The solution process will result in a quadratic equation that can be solved to find the value of <InlineMath math="v" />.

## Common Misconceptions About Quadratic Equations

Some common misconceptions include:

1. Identifying the addition operation <InlineMath math="x + 3" /> as <InlineMath math="3x" />.

   **Concrete example**: If a room's length is <InlineMath math="x" /> meters, and increases by 3 meters, then its length becomes <InlineMath math="x + 3" /> meters, not <InlineMath math="3x" /> meters.

2. Labeling an equation as a quadratic equation simply because the highest power of the variable <InlineMath math="x" /> is 2, without considering the overall form of the equation.

   Remember that a quadratic equation is a polynomial with the standard form <InlineMath math="ax^2 + bx + c = 0" /> where <InlineMath math="a \neq 0" />.

## Forms of Quadratic Equations

Consider the following forms, which ones are quadratic equations?

1. <InlineMath math="\frac{1}{x} + 2x + 4 = 0" />

   This is not a quadratic equation because it contains the term <InlineMath math="\frac{1}{x}" />.

2. <InlineMath math="\frac{1}{x-5} + \frac{1}{x-3} = x^2 - 4" />

   This is not a quadratic equation in standard form, because it has a fractional form with variables in the denominator.

3. <InlineMath math="3x^2 + 2x - 1 = 0" />

   This is a quadratic equation because it is in the form <InlineMath math="ax^2 + bx + c = 0" /> with <InlineMath math="a = 3 \neq 0" />.

4. <InlineMath math="x^2 + \frac{1}{x} + 3 = 0" />

   This is not a quadratic equation because it contains the term <InlineMath math="\frac{1}{x}" />.

## Practice Problems

**Identifying Quadratic Equations**

Determine whether the following mathematical equations are quadratic equations:

1. <InlineMath math="x^3 + x^2 + x = 0" />
2. <InlineMath math="3x^2 + 2x - 1 = 0" />
3. <InlineMath math="\frac{1}{x} + 5x = 0" />
4. <InlineMath math="x^2 + \frac{1}{x} + 3 = 0" />

**Factorization**

Expand the following equations:

1. <InlineMath math="(x + 2)(x + 3) = 0" />
2. <InlineMath math="\left(\frac{1}{3}x - 4\right)(x + 9) = 0" />
3. <InlineMath math="(2x - 8)(x + 5) = 0" />

### Answer Key

**Identifying Quadratic Equations**

1. <InlineMath math="x^3 + x^2 + x = 0" />

   **Answer**: Not a quadratic equation, because it has the highest power of 3 (<InlineMath math="x^3" />). This is a cubic equation.

2. <InlineMath math="3x^2 + 2x - 1 = 0" />

   **Answer**: Quadratic equation, because it is in the form <InlineMath math="ax^2 + bx + c = 0" /> with <InlineMath math="a = 3" />, <InlineMath math="b = 2" />, and <InlineMath math="c = -1" />.

3. <InlineMath math="\frac{1}{x} + 5x = 0" />

   **Answer**: Not a quadratic equation, because it contains the term <InlineMath math="\frac{1}{x}" />. This is a fractional equation.

4. <InlineMath math="x^2 + \frac{1}{x} + 3 = 0" />

   **Answer**: Not a quadratic equation, because it contains the term <InlineMath math="\frac{1}{x}" />. This is a mixed equation.

**Factorization**

1. <InlineMath math="(x + 2)(x + 3) = 0" />

   **Answer**:

   <div className="flex flex-col gap-4">
     <BlockMath math="(x + 2)(x + 3) = 0" />
     <BlockMath math="x^2 + 3x + 2x + 6 = 0" />
     <BlockMath math="x^2 + 5x + 6 = 0" />
   </div>

   So, the product of the two factors is <InlineMath math="x^2 + 5x + 6 = 0" />

2. <InlineMath math="\left(\frac{1}{3}x - 4\right)(x + 9) = 0" />

   **Answer**:

   <div className="flex flex-col gap-4">
     <BlockMath math="\left(\frac{1}{3}x - 4\right)(x + 9) = 0" />
     <BlockMath math="\frac{1}{3}x \cdot x + \frac{1}{3}x \cdot 9 - 4 \cdot x - 4 \cdot 9 = 0" />
     <BlockMath math="\frac{1}{3}x^2 + 3x - 4x - 36 = 0" />
     <BlockMath math="\frac{1}{3}x^2 - x - 36 = 0" />
   </div>

   Multiply all terms by 3 to simplify:

   <div className="flex flex-col gap-4">
     <BlockMath math="x^2 - 3x - 108 = 0" />
   </div>

   So, the product of the two factors is <InlineMath math="x^2 - 3x - 108 = 0" /> or <InlineMath math="\frac{1}{3}x^2 - x - 36 = 0" />

3. <InlineMath math="(2x - 8)(x + 5) = 0" />

   **Answer**:

   <div className="flex flex-col gap-4">
     <BlockMath math="(2x - 8)(x + 5) = 0" />
     <BlockMath math="2x \cdot x + 2x \cdot 5 - 8 \cdot x - 8 \cdot 5 = 0" />
     <BlockMath math="2x^2 + 10x - 8x - 40 = 0" />
     <BlockMath math="2x^2 + 2x - 40 = 0" />
   </div>

   Factoring <InlineMath math="2x - 8" /> as <InlineMath math="2(x - 4)" />:

   <div className="flex flex-col gap-4">
     <BlockMath math="2(x - 4)(x + 5) = 0" />
   </div>

   So, the product of the two factors is <InlineMath math="2x^2 + 2x - 40 = 0" />

**Solving Quadratic Equations**

Let's solve some equations from the factorization results above:

1. <InlineMath math="x^2 + 5x + 6 = 0" />

   **Answer**:
   Factorization:

   <div className="flex flex-col gap-4">
     <BlockMath math="x^2 + 5x + 6 = 0" />
     <BlockMath math="(x + 2)(x + 3) = 0" />
   </div>

   Therefore:

   - If <InlineMath math="x + 2 = 0" />, then <InlineMath math="x = -2" />
   - If <InlineMath math="x + 3 = 0" />, then <InlineMath math="x = -3" />

   The roots of the equation are: <InlineMath math="x = -2" /> or <InlineMath math="x = -3" />

2. <InlineMath math="x^2 - 3x - 108 = 0" />

   **Answer**:
   Using the quadratic formula:

   <div className="flex flex-col gap-4">
     <BlockMath math="x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}" />
     <BlockMath math="x = \frac{3 \pm \sqrt{9 + 432}}{2}" />
     <BlockMath math="x = \frac{3 \pm \sqrt{441}}{2}" />
     <BlockMath math="x = \frac{3 \pm 21}{2}" />
   </div>

   Therefore:

   - <InlineMath math="x = \frac{3 + 21}{2} = 12" />
   - <InlineMath math="x = \frac{3 - 21}{2} = -9" />

   The roots of the equation are: <InlineMath math="x = 12" /> or <InlineMath math="x = -9" />

   Verification by factorization:

   <div className="flex flex-col gap-4">
     <BlockMath math="x^2 - 3x - 108 = 0" />
     <BlockMath math="(x - 12)(x + 9) = 0" />
   </div>

3. <InlineMath math="2x^2 + 2x - 40 = 0" />

   **Answer**:
   Simplify the equation by dividing all terms by 2:

   <div className="flex flex-col gap-4">
     <BlockMath math="x^2 + x - 20 = 0" />
   </div>

   Factorization:

   <div className="flex flex-col gap-4">
     <BlockMath math="x^2 + x - 20 = 0" />
     <BlockMath math="(x + 5)(x - 4) = 0" />
   </div>

   Therefore:

   - If <InlineMath math="x + 5 = 0" />, then <InlineMath math="x = -5" />
   - If <InlineMath math="x - 4 = 0" />, then <InlineMath math="x = 4" />

   The roots of the equation are: <InlineMath math="x = -5" /> or <InlineMath math="x = 4" />
