Question

How do you write \[{x^2} - 25\] in factored form?

Answer 1

Hint: Here in this question, we have to find the factors of the given algebraic expression. To solve this first we have to write 25 as the square number of five then its look similar as a algebraic identity \[\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)\] on using this identity on further simplification, we get the required solution.

Complete step by step solution:
The given equation is the form of an algebraic equation. the equation is represents the algebraic equation \[\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)\] .
Factored form is defined as the simplest algebraic expression in which no common factors remain. Finding the factored form is useful in solving linear equations. Factored form may be a product of greatest common factors or the difference of squares.
Consider the given algebraic expression:
 \[ \Rightarrow {x^2} - 25\] ------(1)
As we know, 25 is the square number of 5 i.e., \[{5^2} = 25\] , then above equation can be written as
 \[ \Rightarrow {x^2} - {5^2}\] -------(2)
Now, its look similar as algebraic identity \[\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right)\] [Algebraic Identities The algebraic equations which are valid for all values of variables in them are called algebraic identities. They are also used for the factorization of polynomials.]
Where a=x and b=5
Then equation (2) becomes:
 \[ \Rightarrow {x^2} - {5^2} = \left( {x - 5} \right)\left( {x + 5} \right)\]
 \[ \Rightarrow \,\,\left( {x - 5} \right)\left( {x + 5} \right)\]
Hence, \[\left( {x - 5} \right)\] and \[\left( {x + 5} \right)\] are the factors of the algebraic equation \[{x^2} - 25\] .
So, the correct answer is “ \[\left( {x - 5} \right)\] and \[\left( {x + 5} \right)\] ”.

Note: The quadratic equation can also be solved by using the factorisation method and we also find the roots by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] . While factorising we use sum product rule, the sum product rule is given as the product factors of the number c is equal to the sum of the factors which satisfies the value of b.