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###  Difficult Problems

1

$2x^2-7x+3=0$
2

Factor the trinomial $2x^2-7x+3$ of the form $ax^2+bx+c$, first, make the product of $2$ and $3$

$\left(2\right)\left(3\right)=6$
3

Now, find two numbers that multiplied give us $6$ and add up to $-7$

$\begin{matrix}\left(-1\right)\left(-6\right)=6\\ \left(-1\right)+\left(-6\right)=-7\end{matrix}$

4

Rewrite the original expression

$2x^2-6x-x+3=0$
5

Factor $2x^2-6x-x+3$ by the greatest common divisor $2$

$2\left(x^2-3x\right)-x+3=0$
6

Factor the polynomial $\left(x^2-3x\right)$ by it's greatest common factor (GCF): $x$

$2x\left(x-3\right)-x+3=0$
7

Factoring by $x-3$

$\left(x-3\right)\left(2x-1\right)=0$
8

Break the equation in $2$ factors and set each equal to zero, to obtain

$x-3=0,\:2x-1=0$
9

Solve the equation ($1$)

$x-3=0$
10

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-3$ from both sides of the equation

$x-3+3=0+3$

Canceling terms on both sides

$x=0+3$

$x+0=x$, where $x$ is any expression

$x=3$
11

Canceling terms on both sides

$x=3$
12

Solve the equation ($2$)

$2x-1=0$
13

We need to isolate the dependent variable $x$, we can do that by simultaneously subtracting $-1$ from both sides of the equation

$2x-1+1=0+1$

Canceling terms on both sides

$2x=0-1\cdot -1$

$x+0=x$, where $x$ is any expression

$2x=-1\cdot -1$

Multiply $-1$ times $-1$

$2x=1$
14

Canceling terms on both sides

$2x=1$
15

Divide both sides of the equation by $2$

$\frac{2x}{2}=\frac{1}{2}$
16

Simplifying the quotients

$x=\frac{1}{2}$
17

Combining all solutions, the $2$ solutions of the equation are

$x=3,\:x=\frac{1}{2}$

$x=3,\:x=\frac{1}{2}$