Quadratic Equations

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:

with the condition that $a \neq 0$ and $a, b, c$ 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 $x^2$ which can be viewed as the area of a square with side length $x$.

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:

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

• If $2x + 1 = 0$, then $x = -\frac{1}{2}$
• If $x - 2 = 0$, then $x = 2$

Therefore, the roots of the quadratic equation are $x = -\frac{1}{2}$ or $x = 2$.

Completing the Square

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

Example:

We divide all terms by 2:

Move the constant to the right side:

Add $\left(\frac{-3/2}{2}\right)^2 = \frac{9}{16}$ to both sides:

Therefore:

Using the Quadratic Formula

For the equation $ax^2 + bx + c = 0$, the roots can be determined using the formula:

Example:

With $a = 2$, $b = -3$, and $c = -2$:

Therefore:

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 $x$ meters, then the remaining area of the room for arranging student seating is:

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 $p$ and $q$, then:

• $p + q = 16$, so $q = 16 - p$
• $p \times q = 63$

Substituting the value of $q$:

By factoring this equation or using the quadratic formula, we can find the values of $p$ and $q$.

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 $v$ km/h and the initial travel time is $t$ hours, then:

• $v \times t = 320$ (distance = speed × time)
• $(v + 24) \times (t - 3) = 320$ (second condition)

From the first equation: $t = \frac{320}{v}$

Substituting into the second equation:

The solution process will result in a quadratic equation that can be solved to find the value of $v$.

Common Misconceptions About Quadratic Equations

Some common misconceptions include:

1. Identifying the addition operation $x + 3$ as $3x$.

   Concrete example: If a room's length is $x$ meters, and increases by 3 meters, then its length becomes $x + 3$ meters, not $3x$ meters.

2. Labeling an equation as a quadratic equation simply because the highest power of the variable $x$ is 2, without considering the overall form of the equation.

   Remember that a quadratic equation is a polynomial with the standard form $ax^2 + bx + c = 0$ where $a \neq 0$.

Forms of Quadratic Equations

Consider the following forms, which ones are quadratic equations?

1. $\frac{1}{x} + 2x + 4 = 0$

   This is not a quadratic equation because it contains the term $\frac{1}{x}$.

2. $\frac{1}{x-5} + \frac{1}{x-3} = x^2 - 4$

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

3. $3x^2 + 2x - 1 = 0$

   This is a quadratic equation because it is in the form $ax^2 + bx + c = 0$ with $a = 3 \neq 0$.

4. $x^2 + \frac{1}{x} + 3 = 0$

   This is not a quadratic equation because it contains the term $\frac{1}{x}$.

Practice Problems

Identifying Quadratic Equations

Determine whether the following mathematical equations are quadratic equations:

1. $x^3 + x^2 + x = 0$
2. $3x^2 + 2x - 1 = 0$
3. $\frac{1}{x} + 5x = 0$
4. $x^2 + \frac{1}{x} + 3 = 0$

Factorization

Expand the following equations:

1. $(x + 2)(x + 3) = 0$
2. $\left(\frac{1}{3}x - 4\right)(x + 9) = 0$
3. $(2x - 8)(x + 5) = 0$

Answer Key

Identifying Quadratic Equations

1. $x^3 + x^2 + x = 0$

   Answer: Not a quadratic equation, because it has the highest power of 3 ($x^3$). This is a cubic equation.

2. $3x^2 + 2x - 1 = 0$

   Answer: Quadratic equation, because it is in the form $ax^2 + bx + c = 0$ with $a = 3$, $b = 2$, and $c = -1$.

3. $\frac{1}{x} + 5x = 0$

   Answer: Not a quadratic equation, because it contains the term $\frac{1}{x}$. This is a fractional equation.

4. $x^2 + \frac{1}{x} + 3 = 0$

   Answer: Not a quadratic equation, because it contains the term $\frac{1}{x}$. This is a mixed equation.

Factorization

1. $(x + 2)(x + 3) = 0$

   Answer:

   $$(x + 2)(x + 3) = 0$$

   $$x^2 + 3x + 2x + 6 = 0$$

   $$x^2 + 5x + 6 = 0$$

   So, the product of the two factors is $x^2 + 5x + 6 = 0$

2. $\left(\frac{1}{3}x - 4\right)(x + 9) = 0$

   Answer:

   $$\left(\frac{1}{3}x - 4\right)(x + 9) = 0$$

   $$\frac{1}{3}x \cdot x + \frac{1}{3}x \cdot 9 - 4 \cdot x - 4 \cdot 9 = 0$$

   $$\frac{1}{3}x^2 + 3x - 4x - 36 = 0$$

   $$\frac{1}{3}x^2 - x - 36 = 0$$

   Multiply all terms by 3 to simplify:

   $$x^2 - 3x - 108 = 0$$

   So, the product of the two factors is $x^2 - 3x - 108 = 0$ or $\frac{1}{3}x^2 - x - 36 = 0$

3. $(2x - 8)(x + 5) = 0$

   Answer:

   $$(2x - 8)(x + 5) = 0$$

   $$2x \cdot x + 2x \cdot 5 - 8 \cdot x - 8 \cdot 5 = 0$$

   $$2x^2 + 10x - 8x - 40 = 0$$

   $$2x^2 + 2x - 40 = 0$$

   Factoring $2x - 8$ as $2(x - 4)$:

   $$2(x - 4)(x + 5) = 0$$

   So, the product of the two factors is $2x^2 + 2x - 40 = 0$

Solving Quadratic Equations

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

1. $x^2 + 5x + 6 = 0$

   Answer: Factorization:

   $$x^2 + 5x + 6 = 0$$

   $$(x + 2)(x + 3) = 0$$

   Therefore:

   • If $x + 2 = 0$, then $x = -2$
   • If $x + 3 = 0$, then $x = -3$

   The roots of the equation are: $x = -2$ or $x = -3$

2. $x^2 - 3x - 108 = 0$

   Answer: Using the quadratic formula:

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

   $$x = \frac{3 \pm \sqrt{9 + 432}}{2}$$

   $$x = \frac{3 \pm \sqrt{441}}{2}$$

   $$x = \frac{3 \pm 21}{2}$$

   Therefore:

   • $x = \frac{3 + 21}{2} = 12$
   • $x = \frac{3 - 21}{2} = -9$

   The roots of the equation are: $x = 12$ or $x = -9$

   Verification by factorization:

   $$x^2 - 3x - 108 = 0$$

   $$(x - 12)(x + 9) = 0$$
3. $2x^2 + 2x - 40 = 0$

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

   $$x^2 + x - 20 = 0$$

   Factorization:

   $$x^2 + x - 20 = 0$$

   $$(x + 5)(x - 4) = 0$$

   Therefore:

   • If $x + 5 = 0$, then $x = -5$
   • If $x - 4 = 0$, then $x = 4$

   The roots of the equation are: $x = -5$ or $x = 4$