How to find out X in a trinomial How can I find out what X equals in this?
$$x^2 - 2x - 3 = 117$$
How would I get started? I'm truly stuck.
 A: Hint: Use the quadratic formula! Rewrite your equation as:
$$x^{2} - 2x - 120 = 0$$
The quadratic formula says that for a quadratic equation of the form $ax^{2} + bx + c = 0$, the roots of the equation (i.e., values of $x$ satisfying the equation), we have:
$$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$$
Using this information, can you solve your equation?
(Note: ordinarily, I would prove the quoted results, but the quadratic formula is a ubiquitous result, information about which can be found just about anywhere on the internet. For proof of the quadratic formula, in particular, 
André Nicolas has provided a particularly nice one here.)
A: Hint:1.$$ax^2 +bx +c=0 \to D=b^2-4ac\ge0 $$$$\ \to\begin{cases}
\color{green}{x_1=\frac{-b+\sqrt{D}}{2a}}  \\
\color{red}{x_2=\frac{-b-\sqrt{D}}{2a}} \\
\end{cases} $$$$$$ 2. $$x^2 +bx +c=(x+\frac{b}{2})^2=\frac{b^2}{4}-c\ge0\quad \text{then} x=\pm\sqrt{\frac{b^2}{4}-c}-\frac{b}{2}$$ 
3. find $x_1$and $x_2 $by solving following  system 
\begin{cases}
x_1+x_2=\frac{-b}{a}   \\
x_1x_2=\frac{c}{a} \\
\end{cases} 
A: As answered by others, you can use the quadratic formula to find the roots of any quadratic polynomial. But this one is a special case where you can simply factor the polynomial.
$$
x^2 - 2x - 3 = 117 \\
x^2 - 2x - 120 = 0 \\
x^2 + 10x - 12x - 120 = 0 \\
x(x + 10) - 12(x + 10) = 0 \\
(x + 10)(x - 12) = 0
$$
So the two solutions are $x = -10$ and $x = 12$.
A: A different way if you have not seen the quadratic formula yet.
Recall that $(x-1)^2 = x^2-2x+1$ and so
$$
x^2-2x-3 = \left(x^2-2x+1\right)-4=(x-1)^2-4
$$
and your equation $120=x^2-2x-3$ becomes equivalent to
$$
117=(x-1)^2-4
$$
so $(x-1)^2 = 121 = 11^2$. Therefore, $x-1 = 11$ or $x-1 = -11$.
In the first case, $x = 11+1 = 12$.
In the second case, $x = -11+1 = -10$.
Check
In the first case, $x=12$ so $x^2-2x-3 = 144-24-3 = 117$.
In the second case, $x=-10$, so $x^2-2x-3 = 100+20-3 = 117$.
