Quadratic Equations

Introduction

Consider the following problem.

A bathroom wall has a rectangular shape and will be covered with square tiles. The length of the wall is 5 tiles longer than its width. If 300 tiles are needed to cover the wall, determine the length of the wall. Assume the area of one tile is 1 square unit.

Problems like this can be modeled using a quadratic equation.

Definition of a Quadratic Equation

A quadratic equation is an equation containing a variable with highest degree two.

The general form is:

where:

• $a$, $b$, and $c$ are real numbers,
• $a$ is the coefficient of $x^2$,
• $b$ is the coefficient of $x$,
• $c$ is a constant,
• $a \neq 0$.

Types of Quadratic Equations

Observe the following examples.

1. $2x^2-6x+5=0$
2. $x^2-4=0$
3. $x^2-9x=0$

Explanation:

1. The equation $2x^2-6x+5=0$ has $a=2$, $b=-6$, and $c=5$. Since all terms are present and nonzero, it is called a complete quadratic equation.
2. The equation $x^2-4=0$ has $a=1$, $b=0$, and $c=-4$. Since it has no $x$ term, it is called a pure quadratic equation.
3. The equation $x^2-9x=0$ has $a=1$, $b=-9$, and $c=0$. Since its constant term is zero, it is called an incomplete quadratic equation.

Some equations are not written directly in general form, for example:

1. $x^2+7x=-10$
2. $\dfrac{1}{x}+\dfrac{1}{x-1}=\dfrac{3}{2}$

Such equations can still be transformed into the general quadratic form using algebraic operations.

Example

Rewrite $2x^2=3x+20$ into general form, then determine $a$, $b$, and $c$.

Therefore:

• $a=2$
• $b=-3$
• $c=-20$

Roots of Quadratic Equations

A value of $x$ that makes the equation true is called a root or solution of the quadratic equation.

In general, the roots can be found by:

1. factoring,
2. completing the square,
3. using the quadratic formula.

1. Factoring

Factoring changes an algebraic expression into a product form.

Example 1

Find the roots of:

Solution:

So:

Thus, the roots are:

Example 2

Find the solution set of:

Solution:

Hence:

So, the solution set is:

Example 3

Find the roots of:

Find two numbers whose sum is $-7$ and whose product is $10$, namely $-5$ and $-2$.

Thus:

So, the roots are:

Related Reading

1. Quadratic Function
2. Solving Quadratic Equations
3. Assessment of Quadratic Equations

2. Completing the Square

Not all quadratic equations are easy to solve by factoring. Another method is completing the square.

Steps:

1. make sure the coefficient of $x^2$ is 1,
2. move the constant to the right side if needed,
3. add the square of half the coefficient of $x$ to both sides,
4. rewrite into perfect-square form,
5. solve using square roots.

Example

Solve:

Solution:

Half of the coefficient of $x$ is:

Its square is:

Add 9 to both sides:

Then:

So:

Therefore, the solution set is:

3. Quadratic Formula

Besides factoring and completing the square, a quadratic equation can be solved using the quadratic formula:

It is derived as follows:

Add the following to both sides:

Then:

The quantity $b^2-4ac$ is called the discriminant, denoted by $D$.

Example

Find the roots of:

Known:

• $a=1$
• $b=-1$
• $c=-6$

Use the quadratic formula:

Therefore:

So, the roots are:

Evaluation

After studying this material, complete the evaluation by following these steps:

1. click Start Quiz,
2. choose the correct answer,
3. click Next for the next question,
4. click Prev to return to the previous question,
5. click Reset to start again,
6. click Submit after finishing all questions,
7. check the score displayed at the bottom,
8. final value equals total score divided by maximum score, then multiplied by 100.

Closing

That concludes this material on quadratic equations for Grade IX SMP/MTs semester 1. If you find any mistakes, please leave a correction in the comments.