Question

What should be subtracted from $4x^{2}-20x+30$ to make it a perfect square?

Answer 1

Hint: In this question it is given that we have to subtract some value from $4x^{2}-20x+30$ to make it a perfect square, so for this we have to write the given quadratic polynomial [quadratic polynomial is a polynomial where the highest power of the variable i.e highest power of x is 2] into $a^{2}-2ab+b^{2}$ form, and as we know that $\left( a^{2}-2ab+b^{2}\right) =\left( a-b\right)^{2}$.

Also we have to know another thing that is any term (y) is called perfect square if we are able to write $y=t^{2}$, where t be a natural number.

Complete step-by-step answer:
given quadratic polynomial,
$4x^{2}-20x+30$ …………………..statement(1)

So we can write equation(1) as,
$2^{2}\times x^{2}-2\times 10\times x+30$
$ \Rightarrow \left( 2x\right)^{2} -2\times 2x\times 5+30$.............statement(2)

 Now if you compare statement(2) with $a^{2}-2ab+b^{2}$ we can easily say that a=2x and b=5, so our statement(2) can be written as
$\left( 2x\right)^{2} -2\times 2x\times 5+5^{2}+5$............statement(3)
{ where we have written 30 as $5^{2}+5$}

So by $\left( a^{2}-2ab+b^{2}\right) =\left( a-b\right)^{2}$ this formula we can write statement(3) as,
$\left( 2x\right)^{2} -2\times 2x\times 5+5^{2}+5$
$\Rightarrow \left( 2x-5\right)^{2} +5$
Now from the above statement we can easily say that if we subtract 5 from the above statement hen the above statement becomes a perfect square, i.e $\Rightarrow \left( 2x-5\right)^{2}$
This is our required solution.

Note: If this type of polynomial is given to you then you have to first identify that which type identity it can be formed and the identities are $\left( a^{2}-2ab+b^{2}\right) =\left( a-b\right)^{2}$ and $\left( a^{2}+2ab+b^{2}\right) =\left( a+b\right)^{2}$