Question

49x² - 25y² + 10y - 1

lesson - factorisation​

Answer 1

The complete question is :

Break down the given expression into factors: $49x^2 - 25y^2+ 10y - 1$

Answer:

The expression  $49x^2 - 25y^2+ 10y - 1$ after factorisation gives the factors

$7x+5y-1$ and $7x-5y+1$ .

Solution:

The given expression is $49x^2 - 25y^2+ 10y - 1$

Factorisations means we have to break the given expression into 2 or more factors, or represent the expression as a product of two factors.

A number, matrix, or polynomial is broken down or decomposed into a product of another entity, or components, which when multiplied together yield the original number, matrix, etc.

let us first take the part:

$- 25y^2+ 10y - 1$

$=- (25y^2- 10y + 1)\\=-[(5y)^2-2\cdot5y\cdot1+(1^2)]\\=-(5y-1)^2$

Here we are using the formula $(a-b)^2=a^2-2ab+b^2$ .

Now the complete expression looks like:

$49x^2-(5y-1)^2$

This can be simplified or broken into factors using the formula:

$a^2-b^2=(a+b)(a-b)$

So the above expression will become:

$49x^2-(5y-1)^2\\=(7x)^2-(5y-1)^2\\=(7x+5y-1)(7x-5y+1)$

Therefore upon successful factorisation, we get the factors:$7x+5y-1$ and $7x-5y+1$ .

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