Question

Factorise the following
$x {}^{2} - 81$
​

Answer 1

Answer:

(x-9)(X+9)

Step-by-step explanation:

x^2-81

(x-9)(X+9)

using Identity (a+b)(a-b)

Answer 2

Given:

• Equation : x² - 81

To Find:

The factorised form of the given equation

Solution:

Now here we have the quadratic equation which is x² - 81 and said that we have to factorise this, as we can't factorise this directly so let's convert it into to suitable simpler forms.

So here ,

we know that 81 is a perfect square which is a square of the number 9

Hence, we simply this into,

$\longrightarrow \tt \: {x}^{2} - {9}^{2}$

Now, let's let's use appropriate algebraic identities to factorise this,

Identity used :

$\star \: \: \: \pink{ \boxed{ \tt{ {a}^{2} - {b}^{2} = a \times b + a - b}}}$

Now let's apply the identity in the given equations where x denotes A and 9 denotes B

$\longrightarrow \tt {x}^{2} - {9}^{2} \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \\ \longrightarrow \tt \: x + 9 \times x - 9 \: \: \\ \\ \\ \longrightarrow \tt \: (x + 9)(x - 9) \:$

• Now let's find the roots of the equation!

Case ( i )

$\longrightarrow \tt \: x + 9 = 0 \\ \\ \\ \longrightarrow \tt \: x = - 9 \: \: \: \:$

Case ( ii )

$\longrightarrow \tt \: x - 9 = 0 \\ \\ \\ \longrightarrow \tt \: x = + 9 \: \: \: \: \:$

• Hence forth the two roots of the quadratic equation are - 9 and + 9

Verification:

Now, let's substitute the values of the roots in the equation to see weather we went right!

Case -----1

$\longrightarrow \tt \: {9}^{2} - 81 = 0 \\ \\ \\ \longrightarrow \tt81 - 81 = 0$

${ \pink { \large{ \sf {hece \: verified \dag}}}}$

Case -----2

$\longrightarrow \tt \: { - 9}^{2} - 81 = 0 \\ \\ \\ \longrightarrow \tt81 - 81 = 0 \: \: \:$

${ \pink{ \large{ \sf{hence \: verifed \dag}}}}$

More to know:

Other algebraic identities!

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

Hope this helps!