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Solving Quadratic Equations by Factoring Worksheet



Find the roots of the quadratic equations by factoring:


1) $ (x+3)(x-2)=0$
2) $ 3(x-7)(x-4)=0$
3) $ (2x+1)(2x-1)=0$
4) $ 64p^2-16p=0$
5) $ 9a^2-100a=0$
6) $ 36y^2-16=0$
7) $ 81x^2-144=0$
8) $ 2b^2-18=0$
9) $ \frac{45}{4}x^2-5=0$
10) $ 100y^2-4=0$
11) $ (3x-6)^2-25=0$
12) $ (x-3)^2-(x+3)^2=0$
13) $ x^2+3x-18=0$
14) $ x^2-15x+56=0$
15) $ 9x^2+9x-4=0$
16) $ x^2+7x+6=0$
17) $ 2x^2-9x-5=0$
18) $ 6x^2-138x+792=0$
19) $ x^2+\frac{3}{4}x-\frac{5}{8}=0$
20) $ \frac{x^2}{2}+\frac{101x}{20}+\frac{1}{2}=0$
21) $ x^2-\frac{5}{2}x-\frac{3}{2}=0$
22) $ x^2-2x-8=0$





Factoring Type ${\rm I}$: (x-a)$\cdot$(x-c)=0


- this is the easiest type, as it's already factored. The roots are a and c: $x\in \{a,c\}$

Example: (x-2)(x+3)=0:

The product is zero when one of the factors is zero, so we must solve:

$\hspace{1cm}$ $(x-2)=0$ $\hspace{2cm}$ OR $\hspace{2cm}$ $(x+3)=0$

The roots are 2 and -3, formally: $x\in \{2,-3\}$

Factoring type ${\rm II}$ : (x-a)(x-c)=d, $\quad d\neq 0 $


"Zero is not on the right side of the equation"

When we have an example as follows:

$5(x-3)(x+2)=2$

obviously, if we plug in 3 or -2, we will get zero, but that is not what is on the right side. We must rewrite the equation in standard form, where zero is on the right side. Then it will make sense to search for the roots (roots turn an expression to zero!) In that example, it means to subtract 2:

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

and solve further.

Factoring type ${\rm III}$ : Grouping


Let's look at an example::

$5x^2\color{red}{-21x}+4=0$

We can rewrite the equation equivalently, which will lead to grouping:

$5x^2\color{red}{-20x-x}+4=0$

Factoring out 5x from the first two terms and $(-1)$ from second two terms:

$\color{red}{5}x(x-4)-\color{red}{1}(x-4)=0$

We got the common term $(x-4)$, which we will factor:

$(x-4)(5x-1)=0$

Our equation was factored out. From that form, we can see that the roots are 4 and $\frac{1}{5}$



Tip 1: Fractions in Quadratic Equations

It is always easier to solve quadratic equation when there are no fractions, so, the tip is to get rid of them first by multiplying by the least common multiple. Unless you want to do harcdore thinking of how to group quadratic equation.

Example

$\frac{x^2}{2}+\frac{101x}{20}+\frac{1}{2}=0$

multiply by 20 on both sides to get rid of fractions:

$10x^2+101x+10=0$

tip: you can split 101x as 100x+x and factor...

Tip 2: to factor use the identity

$a^2-b^2=(a-b)(a+b)$