Answer
the solution set is\[\left\{ -3,7 \right\}\].
Work Step by Step
Step 1:Shift all nonzero terms to left side and obtain zero on the other side.
For this, subtract 21 from both the sides as follows:
\[{{x}^{2}}-4x-21=21-21\]
This implies that,
\[{{x}^{2}}-4x-21=0\]
So, after shifting all nonzero terms to the left side, the above equation becomes: \[{{x}^{2}}-4x-21=0\].
Step 2:Find the factor of the above equation:
Consider the equation\[{{x}^{2}}-4x-21=0\].
Factorize it as follows:
\[\begin{align}
& {{x}^{2}}-4x-21=0 \\
& {{x}^{2}}-7x+3x-21=0 \\
& x\left( x-7 \right)+3\left( x-7 \right)=0 \\
& \left( x+3 \right)\left( x-7 \right)=0
\end{align}\]
Steps 3 and 4: Set each factor equal to zero and solve the resulting equation:
From step 2, \[\left( x+3 \right)\left( x-7 \right)=0\].
By the zero product principal, either \[\left( x+3 \right)=0\]or\[\left( x-7 \right)=0\].
Now \[\left( x+3 \right)=0\] implies that\[x=-3\] and \[\left( x-7 \right)=0\]implies that\[x=7\].
Step5: Check the solution in the original equation.
Check for\[x=-3\]. So consider,
\[\begin{align}
& {{x}^{2}}-4x-21=0 \\
& {{\left( -3 \right)}^{2}}-4\times \left( -3 \right)-21=0 \\
& 9+12-21=0 \\
& 0=0
\end{align}\]
And, check for\[x=7\].
\[\begin{align}
& {{x}^{2}}-4x-21=0 \\
& {{7}^{2}}-4\times 7-21=0 \\
& 49-28-21=0 \\
& 0=0
\end{align}\]
Hence, the solution set is\[\left\{ -3,7 \right\}\].
