<s>[INST] How can we make it easy to solve quadratics without the Quadratic Formula? For example ,
$$5x^2 - 7x-2=0$$
Then, we only need to find is when
$$5x^2-7x=2$$
$$x(5x-7)=2$$
Since that when we get $2-2=0$.

Is there any way we can find them easily without solving quadratics using formula?

I am just getting very enthusiastic in this topic.
 [/INST] Unfortunately, your strategy does not help that much because it is very difficult to solve
$$
x(5x-7)=2
$$
without simply returning to the original equation. This is because there is a $2$ on the RHS. The reason we want to be able to write a quadratic in the form
$$
(x-p)(x-q)=0
$$
is so that we can exploit the zero-product property: if $a \times b=0$, then $a=0$ or $b=0$. Once we have written an equation in the form $(x-p)(x-q)=0$, we can still instantly deduce that $x=p$ or $x=q$. This is something we can't do when we have something like
$$
x(5x-7)=2 \, .
$$

Instead, I would suggest dividing through by $5$, which yields
$$
x^2 - \frac{7}{5}x - \frac{2}{5} = 0 \, .
$$
Then, add $2/5$ to both sides:
$$
x^2 - \frac{7}{5}x = \frac{2}{5} \, \tag{*}\label{*} .
$$
The LHS is almost a perfect square. Note that
$$
\left(x-\frac{7}{10}\right)^2=x^2-\frac{7}{5}x+\frac{49}{100} \, .
$$
If we subtract $49/100$ from this equation, we get
$$
\left(x-\frac{7}{10}\right)^2-\frac{49}{100}=x^2-\frac{7}{5}x \, .
$$
Hence, the equation $\eqref{*}$ can be rewritten as
$$
\left(x-\frac{7}{10}\right)^2-\frac{49}{100}=\frac{2}{5} \, .
$$
Add $49/100$ to both sides:
$$
\left(x-\frac{7}{10}\right)^2 = \frac{2}{5}+\frac{49}{100} = \frac{89}{100} \, .
$$
Take the square root of both sides:
$$
x-\frac{7}{10} = \pm\sqrt{\frac{89}{100}} \, .
$$
Add $7/10$ to both sides:
$$
x = \frac{7}{10} \pm \sqrt{\frac{89}{100}} \, ,
$$
and we are done. This method of solving quadratics is known as completing the square. Unlike factorisation, it can be used to solve any quadratic. In fact, the quadratic formula comes from completing the square on the general quadratic equation
$$
ax^2 + bx + c = 0 \, .
$$
Please let me know if you have any questions.
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