We begin with solving quadratics by using a fundamental result.
The Square Root Property
$$\mbox{If $x^2=c,\,$ then $x=\sqrt{c}$ $\,$OR$\,$ $x=-\sqrt{c}.$}$$
Examples
Solve the equation $w^2=4$.
$$
\begin{array}{lll}
&\displaystyle w^2=4&\mbox{}\\
\implies &\displaystyle w=\sqrt{4} \,\,\,\mbox{or}\,\,\,w=-\sqrt{4}&\mbox{by the Square Root Property}\\
\implies &\displaystyle w=2 \,\,\,\mbox{or}\,\,\,w=-2&\mbox{}\\
\end{array}
$$
Solve the equation $z^2=75$.
$$
\begin{array}{lll}
&\displaystyle z^2=75&\mbox{}\\
\implies &\displaystyle z=\sqrt{75} \,\,\,\mbox{or}\,\,\,z=-\sqrt{75}&\mbox{by the Square Root Property}\\
\implies &\displaystyle z=\sqrt{25\cdot 3} \,\,\,\mbox{or}\,\,\,z=-\sqrt{25\cdot 3}&\mbox{}\\
\implies &\displaystyle z=5\sqrt{3} \,\,\,\mbox{or}\,\,\,z=-5\sqrt{3}&\mbox{}\\
\end{array}
$$
Check
$$ \begin{array}{lll} \displaystyle z^2&\displaystyle=\left(5\sqrt{3}\right)^2 &\mbox{}\\ \displaystyle &\displaystyle=5^2\left(\sqrt{3}\right)^2 &\mbox{}\\ \displaystyle &\displaystyle=25 \cdot 3 &\mbox{}\\ \displaystyle &\displaystyle=75 \,\,\,\checkmark &\mbox{}\\ \end{array} $$ $$ \begin{array}{lll} \displaystyle z^2&\displaystyle=\left(-5\sqrt{3}\right)^2 &\mbox{}\\ \displaystyle &\displaystyle=(-5)^2\left(\sqrt{3}\right)^2 &\mbox{}\\ \displaystyle &\displaystyle=25 \cdot 3 &\mbox{}\\ \displaystyle &\displaystyle=75 \,\,\, \checkmark &\mbox{}\\ \end{array} $$
Check
$$ \begin{array}{lll} \displaystyle z^2&\displaystyle=\left(5\sqrt{3}\right)^2 &\mbox{}\\ \displaystyle &\displaystyle=5^2\left(\sqrt{3}\right)^2 &\mbox{}\\ \displaystyle &\displaystyle=25 \cdot 3 &\mbox{}\\ \displaystyle &\displaystyle=75 \,\,\,\checkmark &\mbox{}\\ \end{array} $$ $$ \begin{array}{lll} \displaystyle z^2&\displaystyle=\left(-5\sqrt{3}\right)^2 &\mbox{}\\ \displaystyle &\displaystyle=(-5)^2\left(\sqrt{3}\right)^2 &\mbox{}\\ \displaystyle &\displaystyle=25 \cdot 3 &\mbox{}\\ \displaystyle &\displaystyle=75 \,\,\, \checkmark &\mbox{}\\ \end{array} $$
Plus or Minus Notation $(\pm)$
Writing the solution to the equation $x^2=c$ as $$x=\sqrt{c}\,\,\,\mbox{or}\,\,\,x=-\sqrt{c}$$ is a bit cumbersome.
Instead, will often write $$x=\pm\sqrt{c}$$ For example, for the above equation $z^2=75,$ we may write the solutions in one fell swoop as $$z=\pm 5\sqrt{3}$$
Awesome! We can now solve basic quadratic equations.
$q^2=18$
$$
\begin{array}{lll}
&\displaystyle q^2=18&\mbox{}\\
\implies &\displaystyle q=\pm \sqrt{18}&\mbox{by the Square Root Property}\\
\implies &\displaystyle q=\pm \sqrt{9\cdot 2}&\mbox{}\\
\implies &\displaystyle q=\pm 3\sqrt{2}&\mbox{}\\
\end{array}
$$
$5n^2=2750$
$$
\begin{array}{lll}
&\displaystyle 5n^2=2750 &\mbox{}\\
\implies &\displaystyle \frac{5n^2}{\color{magenta}{5}}=\frac{2750}{\color{magenta}{5}}&\mbox{}\\
\implies &\displaystyle n^2=550&\\
\implies &\displaystyle n=\pm\sqrt{550}&\mbox{by the Square Root Property}\\
\implies &\displaystyle n=\pm\sqrt{25\cdot 22}&\\
\implies &\displaystyle n=\pm 5\sqrt{22}&\\
\end{array}
$$
$3a^2 - 150 = 0$
$$
\begin{array}{lll}
&\displaystyle 3a^2 - 150 = 0&\mbox{}\\
\implies &\displaystyle 3a^2 - 150 \color{magenta}{+150}= \color{magenta}{+150}&\mbox{}\\
\implies &\displaystyle 3a^2 =150&\mbox{}\\
\implies &\displaystyle \frac{3a^2}{\color{magenta}{3}} =\frac{150}{\color{magenta}{3}}&\mbox{}\\
\implies &\displaystyle a^2 =50&\mbox{}\\
\implies &\displaystyle a =\pm\sqrt{50}&\mbox{by the Square Root Property}\\
\implies &\displaystyle a =\pm\sqrt{25\cdot 2}&\mbox{}\\
\implies &\displaystyle a =\pm 5\sqrt{2}&\mbox{}\\
\end{array}
$$
$x^2+121=0$
$$
\begin{array}{lll}
&\displaystyle x^2+121=0&\mbox{}\\
\implies &\displaystyle x^2+121\color{magenta}{-121}=0\color{magenta}{-121}&\mbox{}\\
\implies &\displaystyle x^2=-121&\mbox{}\\
\implies &\displaystyle x=\pm\sqrt{-121}&\mbox{by the Square Root Property}\\
\implies &\displaystyle x=\pm\sqrt{121(-1)}&\mbox{}\\
\implies &\displaystyle x=\pm\sqrt{121}\sqrt{-1}&\mbox{}\\
\implies &\displaystyle x=\pm 11i&\mbox{}\\
\end{array}
$$
And what is our reward for going to all this trouble to learn how to deal with square roots? ...
Answer: We get to solve more equations! :D
Examples
$7p^2-54=0$
$$
\begin{array}{lll}
&\displaystyle 7p^2-54=0&\mbox{}\\
\implies &\displaystyle 7p^2-54\color{magenta}{+54}=0\color{magenta}{+54}&\mbox{}\\
\implies &\displaystyle 7p^2=54&\mbox{}\\
\implies &\displaystyle \frac{7p^2}{\color{magenta}{7}}=\frac{54}{\color{magenta}{7}}&\mbox{}\\
\implies &\displaystyle p^2=\frac{54}{7}&\mbox{}\\
\implies &\displaystyle p=\pm\sqrt{\frac{54}{7}}&\mbox{by the Square Root Property}\\
\implies &\displaystyle p=\pm\frac{\sqrt{54}}{\sqrt{7}}&\mbox{}\\
\implies &\displaystyle p=\pm\frac{\sqrt{9\cdot 6}}{\sqrt{7}}&\mbox{}\\
\implies &\displaystyle p=\pm\frac{3\sqrt{6}}{\sqrt{7}}&\mbox{}\\
\implies &\displaystyle p=\pm\frac{3\sqrt{6}}{\sqrt{7}}\color{magenta}{\frac{\sqrt{7}}{\sqrt{7}}}&\mbox{}\\
\implies &\displaystyle p=\pm\frac{3\sqrt{42}}{7}&\mbox{}\\
\end{array}
$$
Check
$$ \begin{array}{lll} \displaystyle 7\color{magenta}{p}^2-54&\displaystyle=7\left(\color{magenta}{\pm\frac{3\sqrt{42}}{7}}\right)^2-54 &\mbox{}\\ \displaystyle &\displaystyle=7\cdot \frac{9\cdot 42}{49}-54 &\mbox{}\\ \displaystyle &\displaystyle= \frac{\color{blue}{7}\cdot 9\cdot 6\cdot \color{blue}{7}}{\color{blue}{7}\cdot \color{blue}{7}}-54 &\mbox{}\\ \displaystyle &\displaystyle=9\cdot 6-54 &\mbox{}\\ \displaystyle &\displaystyle=54-54 &\mbox{}\\ \displaystyle &\displaystyle=0\,\,\,\checkmark &\mbox{}\\ \end{array} $$
Check
$$ \begin{array}{lll} \displaystyle 7\color{magenta}{p}^2-54&\displaystyle=7\left(\color{magenta}{\pm\frac{3\sqrt{42}}{7}}\right)^2-54 &\mbox{}\\ \displaystyle &\displaystyle=7\cdot \frac{9\cdot 42}{49}-54 &\mbox{}\\ \displaystyle &\displaystyle= \frac{\color{blue}{7}\cdot 9\cdot 6\cdot \color{blue}{7}}{\color{blue}{7}\cdot \color{blue}{7}}-54 &\mbox{}\\ \displaystyle &\displaystyle=9\cdot 6-54 &\mbox{}\\ \displaystyle &\displaystyle=54-54 &\mbox{}\\ \displaystyle &\displaystyle=0\,\,\,\checkmark &\mbox{}\\ \end{array} $$
$(4t - 5)^2-50=0$
$$
\begin{array}{lll}
&\displaystyle (4t - 5)^2-50=0&\mbox{}\\
\implies &\displaystyle (4t - 5)^2-50\color{magenta}{+50}=0\color{magenta}{+50}&\mbox{}\\
\implies &\displaystyle (4t - 5)^2=50&\mbox{}\\
\implies &\displaystyle 4t - 5=\pm \sqrt{50}&\mbox{by the Square Root Property}\\
\implies &\displaystyle 4t - 5=\pm \sqrt{25\cdot 2}&\mbox{}\\
\implies &\displaystyle 4t - 5=\pm 5\sqrt{2}&\mbox{}\\
\implies &\displaystyle 4t - 5\color{magenta}{+5}=\pm 5\sqrt{2}\color{magenta}{+5}&\mbox{}\\
\implies &\displaystyle 4t =5\pm 5\sqrt{2}&\mbox{}\\
\implies &\displaystyle \frac{4t}{\color{magenta}{4}} =\frac{5\pm 5\sqrt{2}}{\color{magenta}{4}}&\mbox{}\\
\implies &\displaystyle t =\frac{5\pm 5\sqrt{2}}{4}&\mbox{}\\
\implies &\displaystyle t =\frac{5+5\sqrt{2}}{4}\,\,\,\mbox{or}\,\,\,t =\frac{5-5\sqrt{2}}{4}&\mbox{}\\
\implies &\displaystyle t \approx 3.017766953\,\,\,\mbox{or}\,\,\,t \approx -0.517766953&\mbox{}\\
\end{array}
$$
Check
Check answer with a calculator.
Check
Check answer with a calculator.
$(y-5)^2+4=0$
$$
\begin{array}{lll}
&\displaystyle (y-5)^2+4=0&\mbox{}\\
\implies &\displaystyle (y-5)^2+4\color{magenta}{-4}=0\color{magenta}{-4}&\mbox{}\\
\implies &\displaystyle (y-5)^2=-4&\mbox{}\\
\implies &\displaystyle y-5=\pm\sqrt{-4}&\mbox{by the Square Root Property}\\
\implies &\displaystyle y-5=\pm\sqrt{4(-1)}&\mbox{}\\
\implies &\displaystyle y-5=\pm\sqrt{4}\sqrt{-1}&\mbox{}\\
\implies &\displaystyle y-5=\pm 2i&\mbox{}\\
\implies &\displaystyle y-5\color{magenta}{+5}=\pm 2i\color{magenta}{+5}&\mbox{}\\
\implies &\displaystyle y=5\pm 2i&\mbox{}\\
\end{array}
$$
Check $$ \begin{array}{lll} \displaystyle (\color{magenta}{y}-5)^2+4&\displaystyle=(\color{magenta}{5\pm 2i}-5)^2+4 &\mbox{}\\ \displaystyle &\displaystyle=(2i)^2+4 &\mbox{}\\ \displaystyle &\displaystyle=2^2i^2+4 &\mbox{}\\ \displaystyle &\displaystyle=4(-1)+4 &\mbox{}\\ \displaystyle &\displaystyle=-4+4 &\mbox{}\\ \displaystyle &\displaystyle=0\,\,\,\checkmark &\mbox{}\\ \end{array} $$
Check $$ \begin{array}{lll} \displaystyle (\color{magenta}{y}-5)^2+4&\displaystyle=(\color{magenta}{5\pm 2i}-5)^2+4 &\mbox{}\\ \displaystyle &\displaystyle=(2i)^2+4 &\mbox{}\\ \displaystyle &\displaystyle=2^2i^2+4 &\mbox{}\\ \displaystyle &\displaystyle=4(-1)+4 &\mbox{}\\ \displaystyle &\displaystyle=-4+4 &\mbox{}\\ \displaystyle &\displaystyle=0\,\,\,\checkmark &\mbox{}\\ \end{array} $$