Question

Solve the equation $\dfrac{1}{2}{(x - 4)^2} = 8$?

Answer 1

Hint: According to the question we have to solve the given expression which is $\dfrac{1}{2}{(x - 4)^2} = 8$. So, to determine the value of the given quadratic expression or to obtain the value of the x first of all we have to rearrange the terms of the given quadratic expression which can be done by multiplying 2 to the both sides of the given expression. Solve the whole square of the terms of the expression which can be done with the help of the formula which is as mentioned below:

Formula used:
$ \Rightarrow {(a - b)^2} = {a^2} + {b^2} - 2ab................(A)$
Now, add and subtract the terms of the obtained quadratic expression and the terms can be any variable or any integer. Take the terms common which can be taken as the common terms to determine the value of the variable x.

Complete step by step answer:
First of all we have to rearrange the terms of the given quadratic expression which can be done by multiplying 2 to the both sides of the given expression. Hence,
$
   \Rightarrow 2 \times \dfrac{1}{2}{(x - 4)^2} = 2 \times 8 \\
   \Rightarrow {(x - 4)^2} = 16 \\
 $

Solve the whole square of the terms of the expression which can be done with the help of the formula (A) which is as mentioned in the solution hint. Hence,
$ \Rightarrow {x^2} + 16 - 8x = 16$
Step 3: Now, we have to add and subtract the terms of the obtained quadratic expression and the terms can be any variable or any integer. Hence,
$
   \Rightarrow {x^2} + 16 - 8x - 16 = 0 \\
   \Rightarrow {x^2} - 8x = 0 \\
 $

Now, take the terms common which can be taken as the common terms to determine the value of the variable x. Hence,
$ \Rightarrow x(x - 8) = 0$
Now, on solving the expression we can easily obtain the roots/zeros of the expression as obtained just above,
$ \Rightarrow x = 0$ and,
$
   \Rightarrow x - 8 = 0 \\
   \Rightarrow x = 8 \\
 $

Hence, with the help of the formula (A) as mentioned in the solution hint we have determined the value of the given quadratic expression which are $x = 0$ and $x = 8$.

Note:
• To determine the required values or we can say roots/zeros of the quadratic expression it is necessary that we have to rearrange the terms and after that we have to open the whole square of the given expression.
• On solving a quadratic expression only two possible zeros/roots can be obtained which will satisfy the expression means on substituting the roots/zeroes will make the expression 0.