Question

Factorize \[ - 81{x^2} + 180x - 100 = 0\]

Answer 1

Hint: The given expression is a quadratic equation. In order to factorize this expression, we will rewrite the equation by writing the coefficient of $ {x^2} $ and coefficient of $ x $ in terms of perfect square. Then we will use the algebraic identity to resolve the expression. With the help of algebraic identity , we can easily find the factors of the given equation.
Formula Used:
We will use the algebraic identity which can be expressed as:
 $ {\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab $

Complete step-by-step answer:
The given equation\[ - 81{x^2} + 180x - 100 = 0\] is the quadratic equation.
We will multiply the above expression with negative sign in order to get equation in standard quadratic equation form. This can be expressed as:
\[81{x^2} - 180x + 100 = 0\]
We know that $ 81{x^2} $ can be written as $ {\left( {9x} \right)^2} $ , also $ 100 $ can be written as $ {10^2} $ . Also 180 can be written as the product of 2, 10 and $ 9x $ . Hence we can rewrite the above equation in the following way:
\[{\left( {9x} \right)^2} - 2 \times 9x \times 10 + {\left( {10} \right)^2} = 0\]
By using the identity $ {\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab $ we have $ 9x $ for $ a $ and $ 10 $ for $ b $ . Hence, we can rewrite the above expression as
\[\begin{array}{l}
{\left( {9x} \right)^2} - 2 \times 9x \times 10 + {\left( {10} \right)^2} = 0\\
{\left( {9x - 10} \right)^2} = 0\\
\left( {9x - 10} \right)\left( {9x - 10} \right) = 0
\end{array}\]
 Hence, we have two roots of equation as
\[\begin{array}{l}
\left( {9x - 10} \right) = 0\\
x = \dfrac{{10}}{9}
\end{array}\]
And \[\left( {9x - 10} \right)\left( {9x - 10} \right) = 0\]is the factorized form of \[ - 81{x^2} + 180x - 100 = 0\].

Note: In this question, we are rewriting the terms in the form of perfect squares. Hence we should have prior knowledge about the squares and roots of the different numbers. We are also using the algebraic identity in the above question. This type of questions can also be solved by factorization method in which we split the coefficient of $ x $ into two term such a way that the product of these two terms , we get the number which is equals to the product of coefficient of $ {x^2} $ and the constant term. Then we can further solve the expression, by finding common terms .