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A quadratic equation is a second-order polynomial. This means that the highest exponent of x is 2. 

A typical quadratic equation is in the form: 


They are three ways of solving the quadratic equation:

1. Factorization method

2. Completing the square method

3. Quadratic formula method

Today, we shall be solving quadratic equations using the factorization method.

Example 1

Solve $x^2+4x-12=0$ by factorization


By factorization





Now, we equate each factor to zero

$x-2=0$ or $x+6=0$

$x=2$ or $x=-6$

Example 2

Factorize the quadratic equation



By factorization





This translates to:

$5x=3$ or $x=-2$

$x=\frac{3}{5}$ or $x=-2$

In some cases, you need to substitute before solving the equation quadratically.

Here is an example:

Example 3

Solve $x^4+4x^2-12=0$


Let $x^2$ be y



By factorization





This translate to

$y-2=0$ or $y+6=0$

$y=2$ or $y=-6$

Recalled that $x^2$ is equal to y
So, when $x^2=2$ $$\sqrt{x^2}=\sqrt{2}$$

Note: the square of a whole number cannot be negative, thus, $-6$ if extraneous
In other cases(as you will see below), the quadratic equation may be re-arranged.

Example 4

Solve $-25=4x^2+20x$ by factorization




Solving quadratically





This translate to


Divide both sides by 2

$\frac{\require{cancel}\bcancel{2}x}{ \require{cancel}\bcancel{2}}=\frac{-5}{2}$

In some cases, you may be given an incomplete quadratic equation
The next example will illustrate an incomplete quadratic equation

Example 5
Solve $4x^2-25=0$ by factorization


This is a difference of two states $(2x)^2$, $(5)$, hence


Hence, either

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

$2x=-5$   or $2x=5$

This translate to

$x=\frac{-5}{2}$ or $x=\frac{5}{2}$
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Example 6

Find the solution to $x^2+\frac{14x}{4}+\frac{49}{16}=0$

First, let take L.C.M



Cross multiply



By factorization




This translates to



Example 7

Solve this: $9x^2+18x+19=26$


First, let's rearrange



By factorization




$3x+7=0$ or $3x-1=0$

$3x=-7$ or $3x=1$

$\frac{\require{cancel}\bcancel{3}x}{\require{cancel}\bcancel{3}}=\frac{-7}{3}$ or  $\frac{\require{cancel}\bcancel{3}x}{\require{cancel}\bcancel{3}}=\frac{1}{3}$ 

$x=\frac{-7}{3}$ or $\frac{1}{3}$.

There you have it! You should be able to solve the quadratic equation by now. If not, then you can always go back to the examples above. 

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