How do you split the middle term in quadratic equations?

Answer
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Hint: In this question, we need to explain how to split the middle term in quadratic equations. A quadratic equation is nothing but it is built up with integers, constants, variables. Mathematically, a symbol (letter) which doesn’t have a value is called a variable. Similarly when it has a fixed value it is called constant. To split the middle term of the quadratic equation, we use the sum product form. The factors of the equation must be in binomial parenthesis. In order to explain how to split the middle term, we can consider a quadratic equation.

Complete step-by-step solution:
Splitting of the middle term :
One of the methods used in factoring the quadratic equation is known as the splitting of terms. When we can’t square the first and the last term of the given quadratic expression then we can use another method to factorise the expression using the splitting of the middle term. Splitting of the middle term is that we have to rewrite the middle term of the quadratic expression as the sum or difference of the two terms. We have to split the middle term into two parts in terms of sum or difference of the terms.
Let us consider a quadratic equation,
\[4x^{2} + \ 8x\ -5\]
Let us consider the quadratic equation as \[f(x)\],
\[f\left( x \right) = {4x}^{2} + \ 8x\ - 5\]
For splitting the middle term of the quadratic equation, we need to consider the middle term of the considered quadratic equation.
Middle term : \[8x\]
In order to split the middle term , we use the sum and product form.
Sum product form :
The general form of the polynomial is \[{ax}^{2}+\ bx\ +c=0\]
The sum product form is the product of the middle term after splitting must be equal to \[a\times\ c\] and the sum must be equal to \[b\] .
\[f\left( x \right) = {4x}^{2} + \ 8x\ - 5\]
Here \[a=4 ,b=8\] and \[c=-5\]
We need to split \[8x\] into two parts,
\[8x = 10x – 2x\]
The product of the terms is \[4×(-5)\] is \[-20\]
The sum of the terms is \[8\]
Now we can rewrite \[8x\] as \[(10x – 2x)\]
Thus we get \[f\left( x \right) = {4x}^{2} + (10x – 2x)- 5\]
Then we can easily factorise the quadratic equation.
Final answer :
We can split the middle term of the quadratic expression as the sum or difference of the two terms.

Note: With the help of the splitting method, we can convert the quadratic expression to algebraic expression. An algebraic expression is nothing but it is built up with integers, constants, variables and mathematical operations (addition, subtraction, multiplication, division etc… ) In mathematics, a symbol (letter) which doesn’t have a value is called a variable. Similarly when it has a fixed value it is called constant. An algebraic expression is nothing but it is a product of two simpler linear expressions.
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