Question

How do you factor ${x^4} + 7{x^2} + 10 = 0$?

Answer 1

Hint: The above given polynomial will have 4 values of the equation by the method of factorization as the power of the variable of the quadratic polynomial is 4.
The given polynomial is called the quadratic polynomial with degree 4.
Above the equation we will solve by using splitting the middle term method.

Complete step-by-step solution:
To factorize the above given quadratic equation we have two methods to solve one is the standard formula method and splitting the middle term, in this problem we will apply splitting the middle term method to factorize the quadratic equation.
$ \Rightarrow {x^4} + 7x + 10 = 0$ ...........1
Let $x^2$= $p$,
$ \Rightarrow {p^2} + 7p + 10 = 0$ (Then equation 1 will become)
Now we will find two numbers which in addition will give 7 and on multiplication it will give 10.
We will check which two numbers will make the product as 10.
5 and 2 , 1 and 10 are the numbers which on multiplication gives 10.
Only 5 and 2 will add up to give 7.
Thus, our equation will become as:
$ \Rightarrow {p^2} + 5p + 2p + 10 = 0$
On taking the common values out;
$ \Rightarrow p(p + 5) + 2(p + 5) = 0$
$ \Rightarrow (p + 2)(p + 5) = 0$
$ \Rightarrow p = - 2,p = - 5$
But, we had assumed $x^2$= $p$
Therefore,
${x^2} = - 2$ and ${x^2} = - 5$ , after removing the square we have;
$x = \pm i\sqrt 2 $,$x = \pm i\sqrt 5 $ (final imaginary values of the quadratic equation given to us)

Four different values are: $x = i\sqrt {2,} - i\sqrt 2 ,i\sqrt 5 , - i\sqrt 5 $

Note: There is another method of making the factors of quadratic equation which is the standard one;
$\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ , where a is the coefficient of x2, b is the coefficient of x and c is the constant term. The above formula gives the two factors as the formula contains two signs plus and minus.