Question

Write these as a product

Answer 1

Answer

a)

${(2b - 5)}^{2} - 36 =( 2b - 11)(2b + 1)$

b)

$9 - {(7 + 3a)}^{2} = (3a - 4)(3a + 11)$

c)

$( {4 - 11m)}^{2} - 1 =( 3 - 11m )( 5 - 11m )$

Explanation

a) The given expresion is

${(2b - 5)}^{2} - 36$

We rewrite as difference of two squares

${(2b - 5)}^{2} - 36 = {(2b - 5)}^{2} - {6}^{2}$

Recall that:

${x}^{2} - {y}^{2} = (x + y)(x - y)$

This implies that:

${(2b - 5)}^{2} - 36 =( {(2b - 5)} -6)(2b - 5 )+ 6)$

Or

${(2b - 5)}^{2} - 36 =( 2b - 5-6)(2b - 5 + 6)$

This simplifies to give:

${(2b - 5)}^{2} - 36 =( 2b - 11)(2b + 1)$

b) The second expression is

$9 - {(7 + 3a)}^{2}$

We rewrite as perfect squares yo get:

$9 - {(7 + 3a)}^{2} = {3}^{2} - {(7 + 3a)}^{2}$

This gives:

$9 - {(7 + 3a)}^{2} = ({3} - {(7 + 3a)})({3} + {(7 + 3a)})$

This implies that

$9 - {(7 + 3a)}^{2} = ({3} - 7 + 3a)({3} + 7 + 3a)$

We simplify to get:

$9 - {(7 + 3a)}^{2} = (3a - 4)(3a + 11)$

c) The third expression is:

$( {4 - 11m)}^{2} - 1$

We obtain the difference of two squares as:

$( {4 - 11m)}^{2} - 1 =( ( {4 - 11m)} - 1 )( ( {4 - 11m)} + 1 )$

We simplify within the parenthesis to get:

$( {4 - 11m)}^{2} - 1 =( 4 - 11m - 1 )( 4 - 11m+ 1 )$

We simplify further to get;

$( {4 - 11m)}^{2} - 1 =( 3 - 11m )( 5 - 11m )$