Question

How do you factor \[32 - 18{x^2}\] ?

Answer 1

Hint: Given question is quadratic form. So we can give an approach to solve it by quadratic formula and get the roots or factors of the equation above. We can also solve this by a simple approach of transposing one of the terms on the other side and cancelling them by 2 will give a ratio of perfect squares. Then taking the square root on both sides gives us the factors.

Complete step-by-step answer:
Method 1: using quadratic formula
Formula used:
Quadratic formula: \[ \Rightarrow \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
given the equation is \[32 - 18{x^2}\]
equating this equation to zero we get,
 \[32 - 18{x^2} = 0\]
Multiplying the whole equation by minus sign and rearranging the terms we get,
 \[18{x^2} - 32 = 0\]
Comparing this with general quadratic equation we get a=18, b=0 and c=-32
The roots can be found using the quadratic formula mentioned above,
 \[ \Rightarrow \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
Putting the values,
 \[ \Rightarrow \dfrac{{ - 0 \pm \sqrt {0 - 4 \times 18 \times \left( { - 32} \right)} }}{{2 \times 18}}\]
Rearranging the terms we get,
 \[ \Rightarrow \dfrac{{ \pm \sqrt {4 \times 18 \times 32} }}{{2 \times 18}}\]
Now we can write 18 as product of 2 and 9, so that we can take the perfect square numbers out of the root
 \[ \Rightarrow \dfrac{{ \pm \sqrt {4 \times 9 \times 2 \times 32} }}{{2 \times 18}}\]
Taking perfect square numbers outside the root we get,
 \[ \Rightarrow \dfrac{{ \pm 2 \times 3 \times 8}}{{2 \times 18}}\]
Cancelling 2 from numerator and denominator, then dividing the ratio of 8 and 18 by 2 we get,
 \[ \Rightarrow \dfrac{{ \pm 3 \times 4}}{9}\]
Further dividing by 3 we get,
 \[ \Rightarrow \dfrac{{ \pm 4}}{3}\]
There we go these are the factors of the given equation above.
So, the correct answer is “ \[ \dfrac{{ \pm 4}}{3}\]”.

Note: Method 2:
In this method note that we will not use quadratic formula but just simple transposing will be used.
Given that \[32 - 18{x^2} = 0\]
taking variable term on other side we get,
 \[18{x^2} = 32\]
Dividing both sides by 2 we get,
 \[9{x^2} = 16\]
Taking the ratio of the numbers we get,
 \[ \Rightarrow {x^2} = \dfrac{{16}}{9}\]
Writing it in square form,
 \[ \Rightarrow {x^2} = {\left( {\dfrac{4}{3}} \right)^2}\]
Taking square root on both sides we get,
 \[ \Rightarrow x = \pm \dfrac{4}{3}\]
This is the correct answer again.