Question

How do you factor completely \[16{x^2} - 8x + 1\]?

Answer 1

Hint: The given equation is a quadratic equation. Here, in this question to find a factors by different methods firstly by using the method of factorisation and also find by comparing the algebraic formula \[{(a - b)^2} = {a^2} - 2ab + {b^2}\]and on further we get the required solution or factors.

Complete step-by-step solution:
Method: 1
Factorization means the process of creating a list of factors otherwise in mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original.

The general form of an equation is \[a{x^2} + bx + c\] when the coefficient of \[{x^2}\] is unity. Every quadratic equation can be expressed as \[{x^2} + bx + c = \left( {x + d} \right)\left( {x + e} \right)\]. Here, b is the sum of d and e and c is the product of d and e.
Consider the given expression \[16{x^2} - 8x + 1\].
consider the factors of the product ac which sum to b, Product of factors of 16 are \[16 = - 4 \times - 4\], \[4 \times 4\] Here, the possible pair of factor gives a summation of -8 is \[\left( { - 4, - 4} \right)\].
Now, Break the middle term as the summation of two numbers such that its product is equal to 16. Calculated above such two numbers are -4 and -4.
\[ \Rightarrow \,\,\,16{x^2} - 4x - 4x + 1\]
Making pairs of terms in the above expression
\[ \Rightarrow \,\,\,\left( {16{x^2} - 4x} \right) - \left( {4x - 1} \right)\]
Take out greatest common divisor GCD from the both pairs, then
\[ \Rightarrow \,\,\,4x\left( {4x - 1} \right) - 1\left( {4x - 1} \right)\]
Take \[\left( {4x - 1} \right)\] common
\[ \Rightarrow \,\,\,\left( {4x - 1} \right)\,\left( {4x - 1} \right)\]
\[ \Rightarrow \,\,\,\left( {4x - 1} \right){\,^2}\]

Method: 2
Consider the given expression \[16{x^2} - 8x + 1\].
It can be written as
\[ \Rightarrow \,\,\,\,{\left( {4x} \right)^2} - 2\left( {4x} \right)\left( 1 \right) + {1^2}\]
Compare the above equation with algebraic formula \[{(a - b)^2} = {a^2} - 2ab + {b^2}\]
Therefore, a=4x and b=1
 Hence we can write as
\[ \Rightarrow \,\,\,\,{\left( {4x - 1} \right)^2}\]
Hence, the factors of the expression \[16{x^2} - 8x + 1\] is \[{\left( {4x - 1} \right)^2}\].


Note: The equation is a quadratic equation. This problem can be solved by using the sum product rule. This defines as for the general quadratic equation \[a{x^2} + bx + c\], the product of \[a{x^2}\] and c is equal to the sum of bx of the equation. Hence we obtain the factors. The factors for the equation depend on the degree of the equation.