Question

How do you know if \[100{{a}^{2}}+100a+{{25}^{{}}}\] is a perfect square trinomial and how do you factor it?

Answer 1

Hint:In the given question, to find the perfect square trinomial we need to first find out the square root of the first term and the third term of the given equation and then you need to twice the product of the two square root, we would find earlier. If the resultant answer is equal to the middle term of the quadratic equation, then it is a perfect square trinomial. For factorization, you need to simplify the equation by performing basic mathematical operations such as addition, subtraction, multiplication and division.

Complete step by step answer:
For a perfect square trinomial, we need to check the first term and the third term of a quadratic equation should be a perfect square. And then check whether the middle term is twice the product of the two square roots. Given quadratic equation:
\[100{{a}^{2}}+100a+{{25}^{{}}}\]
Here,
First term=\[100{{a}^{2}}\]
\[\sqrt{100{{a}^{2}}}=10a\]
Third term=\[25\]
\[\sqrt{25}=5\]
Now, product of the square root of first term and the third term,, we get
\[10a\times 5=50a\]
Doubled the product, we get
\[2\times 50a=100a\]
\[100a\] is the middle term of the quadratic equation.
Therefore, the given equation is the perfect square trinomial.

The binomial is consists of:
Square root of the first term (10a).
The sign of the second term (+)
The square root of the third term (5).
\[{{\left( 10a+5 \right)}^{2}}\] is the required squared binomial.
We have the given quadratic equation:
\[100{{a}^{2}}+100a+{{25}^{{}}}\]
Taking 25 as a common factor from the given quadratic equation, we get
\[25\left( 4{{a}^{2}}+4a+1 \right)\]

Using the sum product pattern we get
\[25\left( 4{{a}^{2}}+2a+2a+1 \right)\]
Taking common factors from the two pairs, we obtain
\[25\left( 2a\left( 2a+1 \right)+1\left( 2a+1 \right) \right)\]
Rewrite the following in a factored form, we get
\[25\left( \left( 2a+1 \right)+\left( 2a+1 \right) \right)\]
\[\Rightarrow 25\times \left( 2a+1 \right)\times \left( 2a+1 \right)\]
Combine the following equation to a square, we get
\[\therefore \]\[25{{\left( 2a+1 \right)}^{2}}\]

Note:A perfect square binomial is a trinomial which when factored gives us the square of a binomial, remember that ‘’trinomial’’ means a ‘’three-term-polynomial’’. Perfect square trinomial are of the form: \[{{a}^{2}}{{x}^{2}}+2axb+{{b}^{2}}\], which are further expressed in squared-binomial form.For factorization, you need to know about the use of the basic mathematical operation and you need to know about the prime factorization of a number.