Question

Please factorize the following :-

25x2-y2+5x+y

Answer 1

Step-by-step explanation:

25x2-y2/5x-y

Final result :

125x2 - xy2 - 5y

————————————————

5

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "y2" was replaced by "y^2". 1 more similar replacement(s).

Step by step solution :

Step 1 :

y2

Simplify ——

5

Equation at the end of step 1 :

y2

((25 • (x2)) - (—— • x)) - y

5

Step 2 :

Equation at the end of step 2 :

xy2

(52x2 - ———) - y

5

Step 3 :

Rewriting the whole as an Equivalent Fraction :

3.1 Subtracting a fraction from a whole

Rewrite the whole as a fraction using 5 as the denominator :

52x2 52x2 • 5

52x2 = ———— = ————————

1 5

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

3.2 Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

52x2 • 5 - (xy2) 125x2 - xy2

———————————————— = ———————————

5 5

Equation at the end of step 3 :

(125x2 - xy2)

————————————— - y

5

Step 4 :

Rewriting the whole as an Equivalent Fraction :

4.1 Subtracting a whole from a fraction

Rewrite the whole as a fraction using 5 as the denominator :

y y • 5

y = — = —————

1 5

Step 5 :

Pulling out like terms :

5.1 Pull out like factors :

125x2 - xy2 = x • (125x - y2)

Trying to factor as a Difference of Squares :

5.2 Factoring: 125x - y2

Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =

A2 - AB + BA - B2 =

A2 - AB + AB - B2 =

A2 - B2

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check : 125 is not a square !!

Ruling : Binomial can not be factored as the

difference of two perfect squares

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Answer 2

Answer: (5x + y) (5x - y + 1)

Step-by-step explanation:

25x^2 - y^2 + 5x + y

By using formula (a^2 - b^2) = (a + b) (a - b) :

= (5 x + y) (5x - y) + 5x + y

= [(5 x + y) (5x - y) + (5x + y) (1)] --------- Since 1 x (5x + y) = 5x + y.

= (5x + y) (5x - y +1) --------- Taking (5x + y) as common from both terms.

∴  25x^2 - y^2 + 5x + y  =  (5x + y) (5x - y + 1)