Quadratic Equations

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, ${{x}^{2}}-6x+5=0$ is the quadratic equation, expressed in the general form of $a{{x}^{2}}+bx+c=0$ , 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 ${{x}^{2}}-6x+5=0$ ?
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 ${{x}^{2}}+7x+12=0$ 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 $10x-3=8x^2$ using factorization.
Answer: Rearrange the equation in the form of $a{{x}^{2}}+bx+c=0$

So we will get (4x – 3)(2x – 1)=0,
4x – 3 = 0 or 2x – 1 = 0
x = $\frac{3}{4}$ or x = $\frac{1}{2}$

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 $a{{x}^{2}}+bx+c=0$ are too large and hard to factorized.

Example 5: Solve $(x-2)=6x(x+3)$ 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
x²-2x-x+2=0
x²-3x+2=0

So the quadratic equation is x² – 3x + 2=0. This is the most basic technique to form up a quadratic equation.

Let’s assume we have the roots of $\alpha$ and $\beta$ :

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

${{x}^{2}}-(\text{sum of roots)}x+(\text{product of roots)}=0$

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

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 $\alpha$ and $\beta$ are the roots of $5{{x}^{2}}-2x-2=0$ , form a quadratic equation $a{{x}^{2}}+bx+c=0$ with the roots of ( $\alpha$ – 5 ) and ( $\beta$ – 5 ).

3) Determine the conditions for the type of roots

Refer back to example 2, we know that ${{x}^{2}}+7x+12=0$ has two different roots (-3 and -4) by solving using factorization method. However, how are we going to determine the types of roots of ${{x}^{2}}+7x+12=0$ without solving the equation? The trick is we can use ${{b}^{2}}-4ac$ .

${{b}^{2}}-4ac$ 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, ${{x}^{2}}+7x+12=0$ , $a=1,b=7,c=12,\text{ so }{{7}^{2}}-4(1)(12)=1$ , we have 2 different roots since the discriminant is greater than zero. Refer table below.

Example 7: A quadratic equation ${{x}^{2}}+2hx+4=x$ 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.

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