Question

How do you factor the expression $25{x^2}\,\, + \,\,70x\,\, + \,\,49$ ?

Answer 1

Hint: To solve this question, we must first understand the basic techniques of factorization. Then we need to assess parameters of the technique that determine whether we can use that method or not. And you may use a mid-term splitting method if applicable here otherwise you may use the quadratic formula and then only we can conclude the correct answer.

Complete Step by Step answer:
Before we move forward with the solution of this given question, let us first understand some basic concepts:
To apply the mid-term splitting method, the condition that we have to check for is that our discriminant must be a perfect square.
Step 1: Given equation is $25{x^2}\,\, + \,\,70x\,\, + \,\,49$
And here, $a = 25\,,\,\,b = 70\,\,,\,\,c = \,\,49$
Step 2: In this step we will find the discriminant and check whether it is perfect square or not:
Since we know that, $D\,\, = \,\,{b^2}\,\, - \,\,4ac$
$
   \Rightarrow D = \,{\left( {70} \right)^2}\,\, - \,\,4 \times 25 \times 49 \\
   \Rightarrow D = 4900\,\, - \,\,4900 \\
   \Rightarrow D = 0 \\
$
And clearly we can see that the discriminant is a perfect square. And so we can use the mid-term splitting. Since the Determinant is zero, therefore we will have equal roots.
Step 3: We can split the equation in following manner:
$25{x^2}\,\, + \,\,70x\,\, + \,\,49\,\, = \,\,{\left( {5x + 7} \right)^2}\,\, = \,\,\left( {5x + 7} \right)\left( {5x + 7} \right)$
And we got our required answer.

Note: Factorization of Quadratic Equation using splitting of the middle term: In this method we split the middle term into two factors. In Factorization of Quadratic Equation using splitting of the middle term which is x term is the sum of two factors and product equal to last term. And if this method is found to be not applicable, then we use the quadratic formula to find the answer.