Question

How do you solve ${x^2} - 8x + 3 = 0$ by completing the square?

Answer 1

Hint: According to the question we have to determine the solution or we can say that the roots of the quadratic expression which can be determined with the help of the making the given quadratic expression in the form of the whole square. So, first of all to make the given quadratic expression in the form of complete square we have compare the given expression with the expression ${x^2} + 2ax + {a^2}$ and then we have to obtain the required term form $2ax$ and ${a^2}$.
Now, we have to add and subtract the term obtained with the help of comparing the expression with $2ax$ and ${a^2}$.
Now, to convert in the form of whole square we have to use the formula which is as mentioned below:

Formula used:
$ \Rightarrow {(a - b)^2} = {a^2} + {b^2} - 2ab.................(A)$
Now, we have to take the square root in the both sides of the equation obtained to determine the roots of the quadratic expression.

Complete step-by-step answer:
Step 1: First of all to make the given quadratic expression in the form of complete square we have compare the given expression with the expression ${x^2} + 2ax + {a^2}$ and then we have to obtain the required term form $2ax$ and ${a^2}$. Hence,
$ \Rightarrow 2ax = - 8x$ and,
$ \Rightarrow {a^2} = 16$
Step 2: Now, with the help of the solution step we can determine the required number to which we have to add and subtract in the quadratic expression to obtain the complete square of the expression. Hence,
$ \Rightarrow $The number is 13.
Step 3: Now, we have to add and subtract the term obtained with the help of comparing the expression with $2ax$ and ${a^2}$. Hence,
$ \Rightarrow {x^2} - 8x + 3 + 13 - 13 = 0$
Step 4: Now, to convert in the form of the whole square we have to use the formula (A) which is as mentioned in the solution hint. Hence,
\[
   \Rightarrow {x^2} - 8x + 16 = 13 \\
   \Rightarrow {(x - 4)^2} = 13 \\
 \]
Step 5: Now, we have to take the square root in the both sides of the equation obtained to determine the roots of the quadratic expression. Hence,
$
   \Rightarrow x - 4 = \pm \sqrt {13} \\
   \Rightarrow x = 4 \pm \sqrt {13} \\
 $

Hence, with the help of the formula (A) we have determined the solution of the given quadratic expression ${x^2} - 8x + 3 = 0$ by completing the square is $ \Rightarrow x = 4 \pm \sqrt {13} $.

Note:
To obtain the both of the roots or the solution of the given quadratic expression we have to compare the expression with ${x^2} + 2ax + {a^2}$ so that, we can determine the term that should be added and subtracted to given expression to convert it in the form of the complete square.
On solving the quadratic expression only two of the roots are possible or we can say that the two zeroes are possible which will satisfy the given quadratic expression.