Question

How do you solve \[{{x}^{2}}-4x-8=0\]?

Answer 1

Hint: The given equation is a quadratic equation, which can be solved in different methods. Here we have to solve and find the roots of the equation \[{{x}^{2}}-4x-8=0\]. But we have to know that not all equations can be factorized, so we have to solve it in some other methods. From the given quadratic equation, we have to find the complete square equation by adding or subtracting values to the given equation and solve it to get the roots of the equation.

Complete step by step answer:
We know that the given quadratic equation is,
\[{{x}^{2}}-4x-8=0\] ……. (1)
We also know that this cannot be directly factorized to get the x values.
So, we have to use another method, that is, completing the square.
Now we can write the equation (1) as,
\[{{x}^{2}}-4x=8\]
Here we can add 4 on both sides to get a complete square equation.
We get,
\[\begin{align}
  & \Rightarrow {{x}^{2}}-4x+4=8+4 \\
 & \Rightarrow {{x}^{2}}-4x+4=12 \\
\end{align}\]
Now, on the left-hand side we have a perfect square equation, so we can replace it with a squared binomial form.
\[\Rightarrow {{\left( x-2 \right)}^{2}}=12\]
Now, we can solve this equation to get the roots of the equation.
We can square on both sides to get,
\[\begin{align}
  & \Rightarrow x-2=\pm \sqrt{12} \\
 & \Rightarrow x-2=\pm \sqrt{4\times 3} \\
 & \Rightarrow x-2=\pm 2\sqrt{3} \\
 & \Rightarrow x=2\pm 2\sqrt{3} \\
\end{align}\]
Therefore, the roots of the equation \[{{x}^{2}}-4x-8=0\] are \[x=2+2\sqrt{3}\] and \[x=2-2\sqrt{3}\].

Note:
 To solve these types of problems we have to know how to solve a quadratic equation by factorization, that’s how we can know whether the given equation can be factorized or not. In these problems, students make mistakes in the complete square by adding or subtracting values in the given equation to get the roots of the equation.