**3 Basic Techniques in Solving Quadratic Equation Questions**

In this chapter we will learn 3 most basic techniques on how to:

**Solve the quadratic equations****Form a quadratic equation****Determine the conditions for the type of roots.**

Generally, is the quadratic equation, expressed in the general form of , where a=1, b=- 6 and c=5. The **root** is the value of x that can solve the equations.

A quadratic equation only has

two roots.

*Example1**: What are the roots of ?
Answer: The value of 1 and 5 are the roots of the quadratic equation, because you will get zero when substitute 1 or 5 in the equation. We will further discuss on how to solve the quadratic equation and find out the roots later.*

**1) ****Solve the quadratic equations**

There are many ways we can use to solve quadratic equations such as using:

1) substitution,

2) inspection,

3) trial and improvement method,

4) **factorization,**

5) **completing the square and**

6) **Quadratic formula.**

However, we will only focus on the **last three methods** as there are the most commonly use methods to solve a quadratic equation in the SPM questions. Let’s move on!

**Factorization**

Factorization is the decomposition of a number into the product of the other numbers, example, 12 could be factored into 3 x 4, 2 x 6, and 1 x 12.

*Example 2**: Solve using factorization.
Answer: We can factor the number 12 into 4 x 3. Remember, always think of the factors which can be added up to the get the middle value (3+4 = 7), refer factorization table below, *

So we will get ( x + 3 )( x + 4 ) = 0,

x + 3 = 0 or x + 4 = 0

x = – 3 or x = – 4

*Example 3**: Solve * *using factorization.
Answer: Rearrange the equation in the form of *

So we will get (4x – 3)(2x – 1)=0,

4x – 3 = 0 or 2x – 1 = 0

x = or x =

**Completing the square **

*Example 4**: Solve the following equation by using completing the square method. *

**Quadratic formula**

Normally when do you need to use this formula?

1) The exam question requested to do so!

2) The quadratic equation **cannot be factorized**.

3) The figure of a, b, and c of the equation are too **large** and **hard to factorized**.

*Example 5**: Solve using quadratic formula.*

**2) ****Form a quadratic equation**

How do you form a quadratic equation if the roots of the equation are 1 and 2? Well, we can do the work out like this using the reverse method:

We can assume:

x = 1 or x = 2

x – 1 = 0 or x – 2 = 0

(x-1)(x-2)=0

x^{2}-2x-x+2=0

x^{2}-3x+2=0

So the quadratic equation is **x ^{2} – 3x + 2=0**. This is the most basic technique to form up a quadratic equation.

Let’s assume we have the roots of and :

In other words, we can form up the equation using the sum of roots (SOR) and product of roots (POR). If the roots are 1 and 2,

SOR = 1+2

= 3

POR = 1 x 2

=2

Sometime we need to determine the SOR and POR from a given quadratic equation in order to find a new equation from a given new roots. In general form,

Let’s look at the example below on how the concept above can help us solve the question.

*Example 6:** Given that and are the roots of , form a quadratic equation with the roots of ( – 5 ) and ( – 5 ). *

**3) ****Determine the conditions for the type of roots**

Refer back to example 2, we know that has two different roots (-3 and -4) by solving using factorization method. However, how are we going to determine the types of roots of without solving the equation? The trick is we can use .

is called a discriminant. Remember, when the value is greater than 0, we have 2 different roots, when it is 0, we have 2 equal roots, and when it is less than 0, we have no roots.

From the quadratic equation, , , we have **2 different roots** since the discriminant is greater than zero. Refer table below.

*Example 7:** A quadratic equation has two equal roots. Find the possible values of h.*

We will look at more SPM questions and example for quadratic equation in next topic on **SPM Questions for Quadratic Equations.**

Further reading

## 2 thoughts on “Quadratic Equations”