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**Hint**: Factorization: Factorization in mathematics is for decomposing or breaking big entities into small pieces. Generally, in factorization when we factorize the number we get all the prime numbers as the factors. It is also called the reverse of the multiplication process.

As we know that ${a^2} - {b^2} = (a + b)(a - b)$

First we try to find a simplified equation of a given equation. First we try to factorize a given algebraic equation. and then we can easily prove the given equation as per output of the algebraic equation.

**Complete step by step solution:**

Given,

\[ = a{b^2} - a{c^2}\]

According to the question.

Taking $a$ common

$ = a({b^2} - {c^2})$

As we know that

$\therefore {a^2} - {b^2} = (a + b)(a - b)$

So

$ = a\{ (b + c)(b - c)\} $

Now

$ \Rightarrow 10{(7.5)^2} - 10{(2.5)^2} = 500$

Taking common 10

$ \Rightarrow 10\{ {(7.5)^2} - {(2.5)^2}\} = 500$

As we know that

$\therefore {a^2} - {b^2} = (a + b)(a - b)$

$ \Rightarrow 10\{ (7.5 + 2.5)(7.5 - 2.5)\} = 500$

$ \Rightarrow 10(10 \times 5) = 500$

$ \Rightarrow 10 \times 50 = 500$

$ \Rightarrow 500 = 500$

\[ \Rightarrow L.H.S = R.H.S\]

Hence proved.

**Note**: Factorization is a method of breaking the expression into products of their factors. There are many factorization identities. To solve the quadratic equation we use the complete square method or directly use the quadratic formula generally known as sridharacharya formula.

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