Question

Solve ${x^2} - 9 = 0.$

Answer 1

Hint:We can use the formula ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$. 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 $x$.

Complete step by step solution:
Given
${x^2} - 9 = 0......................\left( i \right)$
Also ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right).....................\left( {ii} \right)$
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 ${x^2} - 9$.
But we can see that there are no common factors so we can directly express ${x^2} - 9$ according to the identity given above.
Such that:
\[ \Rightarrow {x^2} - 9 = {\left( x \right)^2} - {\left( 3 \right)^2}......................\left( {iii} \right)\]
Now on comparing (i) and (iii) we get:
\[
\Rightarrow {x^2} - 9 = {\left( x \right)^2} - {\left( 3 \right)^2} \\
\Rightarrow {\left( x \right)^2} - {\left( 3 \right)^2} = \left( {x + 3} \right)\left( {x - 3}
\right)..............\left( {iv} \right) \\
\]
Therefore on factorization of ${x^2} - 9$we get\[\left( {x + 3} \right)\left( {x - 3} \right)\].
Now we have to solve\[{x^2} - 9\]:
For that we have to consider equation (i), such that:
$
\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:
\[
\Rightarrow x + 3 = 0\,\,\,\,\,\,{\text{and}}\,\,\,\,\,\,x - 3{\text{ = 0}} \\
\Rightarrow x = - 3\,\,\,\,\,\,{\text{and}}\,\,\,\,\,\,x = 3.............................\left( {vi} \right) \\
\]
Therefore on solving ${x^2} - 9 = 0$ we get \[x = - 3\,{\text{and}}\,x = 3.\]

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