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Question: Solve \({x^2} - 9 = 0.\)...

Solve x2−9=0.{x^2} - 9 = 0.

Explanation

Solution

We can use the formula a2−b2=(a+b)(a−b){a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right). This is the difference of square identity and by using this formula we can factorize the given equation. So in order to solve it we have to convert the question and then express it in the form of the difference of square identity.
Thereby we can solve for xx.

Complete step by step solution:
Given
x2−9=0......................(i){x^2} - 9 = 0......................\left( i \right)
Also a2−b2=(a+b)(a−b).....................(ii){a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right).....................\left( {ii} \right)
So we need to express (ii) in terms of (i):
So we need to express (ii) in terms of (i), for that we have to take the common factors from x2−9{x^2} - 9.
But we can see that there are no common factors so we can directly express x2−9{x^2} - 9 according to the identity given above.
Such that:
⇒x2−9=(x)2−(3)2......................(iii)\Rightarrow {x^2} - 9 = {\left( x \right)^2} - {\left( 3 \right)^2}......................\left( {iii} \right)
Now on comparing (i) and (iii) we get:

⇒x2−9=(x)2−(3)2 ⇒(x)2−(3)2=(x+3)(x−3)..............(iv) \Rightarrow {x^2} - 9 = {\left( x \right)^2} - {\left( 3 \right)^2} \\\ \Rightarrow {\left( x \right)^2} - {\left( 3 \right)^2} = \left( {x + 3} \right)\left( {x - 3} \right)..............\left( {iv} \right) \\\

Therefore on factorization of x2−9{x^2} - 9we get(x+3)(x−3)\left( {x + 3} \right)\left( {x - 3} \right).
Now we have to solvex2−9{x^2} - 9:
For that we have to consider equation (i), such that:
⇒x2−9=0 ⇒(x+3)(x−3)=0...........................(v)  \Rightarrow {x^2} - 9 = 0 \\\ \Rightarrow \left( {x + 3} \right)\left( {x - 3} \right) = 0...........................\left( v \right) \\\
Now equating each part in (v) to zero, we get:

⇒x+3=0      and      x−3 = 0 ⇒x=−3      and      x=3.............................(vi) \Rightarrow x + 3 = 0\,\,\,\,\,\,{\text{and}}\,\,\,\,\,\,x - 3{\text{ = 0}} \\\ \Rightarrow x = - 3\,\,\,\,\,\,{\text{and}}\,\,\,\,\,\,x = 3.............................\left( {vi} \right) \\\

Therefore on solving x2−9=0{x^2} - 9 = 0 we get x=−3 and x=3.x = - 3\,{\text{and}}\,x = 3.

Additional Information:
Another technique for solving x2−9{x^2} - 9 is by using the Quadratic formula which is:
Quadratic Formula: ax2+bx+c=0a{x^2} + bx + c = 0 Here a,  b,  ca,\;b,\;c are numerical coefficients.
So to solve xx we have:x=−b±b2−4ac2ax = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}
So in order to solve the x2−9{x^2} - 9using quadratic formula we have to find the values of
a,  b,  ca,\;b,\;ccorresponding to the given question. Then by substituting the values in the above equation we can find the values for xx and thereby solve it.

Note: Similar questions are to be solved by the above described method since it’s easy to proceed the various steps. While approaching a question one should study it properly and accordingly should choose the method to factorize the polynomial. Polynomial factorization is always done over some set of numbers which may be integers, real numbers or complex numbers.