Question

How do you know if ${x^2} - 24x + 144$ is a perfect square trinomial and how do you factor it?

Answer 1

Hint: We will first write the given polynomial in the form ${a^2} + {b^2} - 2ab$ so that we can use the formula ${(a - b)^2} = {a^2} + {b^2} - 2ab$. Thus, we have a perfect square and factors will be (a – b) twice as well.

Complete step-by-step solution:
We are given that we are required to find out if ${x^2} - 24x + 144$ is a perfect square trinomial and we also need to factorise it.
Let us consider the given quadratic polynomial which is given by the following expression:-
$ \Rightarrow {x^2} - 24x + 144$
We can also write this as follows:-
$ \Rightarrow {x^2} - 2 \times 12 \times x + {12^2}$ …………..(1)
Since, we know that we have a formula given by the following expression:-
$ \Rightarrow {(a - b)^2} = {a^2} + {b^2} - 2ab$
Replacing a by x and b by 12, we will then obtain the following expression as:-
$ \Rightarrow {(x - 12)^2} = {x^2} + {12^2} - 2 \times 12 \times x$
Putting this expression in equation number 1, we will then obtain the following expression as.
The given trinomial can be written as following:-
$ \Rightarrow {(x - 12)^2} = {x^2} - 24x + 144$
Thus, it is a perfect square.
The factors of the given equation ${x^2} - 24x + 144$ are $(x - 12)$ twice.

Note: The students must note that we could have opted to find zeros of the given trinomial and then use them to factorize as well.
Let us assume that the equation is ${x^2} - 24x + 144 = 0$
Now, we can write this expression as follows:-
$ \Rightarrow {x^2} - 12x - 12x + 144 = 0$
Taking x common from the first two terms in the above mentioned expression, we will then obtain the following expression as:-
$ \Rightarrow x\left( {x - 12} \right) - 12x + 144 = 0$
Taking - 12 common from the last two terms in the above mentioned expression, we will then obtain the following expression as:-
$ \Rightarrow x\left( {x - 12} \right) - 12\left( {x - 12} \right) = 0$
Thus, we have the following expression as:-
$ \Rightarrow \left( {x - 12} \right)\left( {x - 12} \right) = 0$
Removing the zero from the right hand side, we will then obtain the following expression as the answer:-
$ \Rightarrow {(x - 12)^2} = {x^2} - 24x + 144$
The students must also know that the term ‘trinomial’ means a polynomial with 3 distinct kinds of terms and the given equation can also be called a quadratic equation.