Question

What do you need to add to complete the square for ${{x}^{2}}+8x$?

Answer 1

Hint: Write $8x$ in the form of $2ab$ by multiplying and dividing it with 2. Now, compare 2 with 2, x with ‘a’ and the one more factor that is present with b. Now, add the square of this factor other than 2 and x and use the algebra identity: - \[{{a}^{2}}+{{b}^{2}}+2ab={{\left( a+b \right)}^{2}}\] to make the given expression a perfect square.

Complete step by step answer:
Here we have been provided with the expression: - ${{x}^{2}}+8x$ and we have been asked to find the missing number or term that must be added so that we can make the expression a perfect square of a binomial.
Now, we know that there are two basic whole square formulas for a binomial expression. They are: - \[{{a}^{2}}+{{b}^{2}}+2ab={{\left( a+b \right)}^{2}}\] and \[{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\]. For the given expression ${{x}^{2}}+8x$ let us assume a = x, we need to find ‘b’. Now, we can say that here we are going to use the formula for \[{{\left( a+b \right)}^{2}}\] because here we get the term $+2ab$ which is analogous to $+8x$, both have positive signs, so we write,
$\Rightarrow {{x}^{2}}+8x={{x}^{2}}+2\times x\times 4$
On comparing $2\times x\times 4$ with $2\times a\times b$ we can conclude that we have b = 4, so we need to add the square of 4 to complete the formula. Therefore we get,
$\Rightarrow {{x}^{2}}+2\times x\times 4+{{4}^{2}}={{\left( x+4 \right)}^{2}}$
Hence, the missing number that must be added is ${{4}^{2}}=16$.

Note: One must remember the two most important and basic algebraic identities given as: - \[{{a}^{2}}+{{b}^{2}}+2ab={{\left( a+b \right)}^{2}}\] and \[{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\]. Remember that the above approach that we have applied to solve the question is known as completing the square method which is a very important part of quadratic equation. The discriminant formula that we used to solve the quadratic equation is derived using the approach of completing the square method. This method is further used in coordinate geometry of parabola for finding the vertex and axis of the parabola.