Step-by-step explanation:
Here is your explanation:
((x4) - 3x2) - 18 = 0 Factoring  x4-3x2-18Â
The first term is,  x4  its coefficient is  1 .
The middle term is,  -3x2  its coefficient is  -3 .
The last term, "the constant", is  -18Â
Step-1 : Multiply the coefficient of the first term by the constant   1 • -18 = -18Â
Step-2 : Find two factors of  -18  whose sum equals the coefficient of the middle term, which is   -3 .
     -18   +   1   =   -17     -9   +   2   =   -7     -6   +   3   =   -3   That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -6  and  3Â
                     x4 - 6x2 + 3x2 - 18
Step-4 : Add up the first 2 terms, pulling out like factors :
                    x2 • (x2-6)
             Add up the last 2 terms, pulling out common factors :
                    3 • (x2-6)
Step-5 : Add up the four terms of step 4 :
                    (x2+3)  •  (x2-6)
             Which is the desired factorization
Find roots (zeroes) of : Â Â Â Â Â Â F(x) = x2+3
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0 Â
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient
In this case, the Leading Coefficient is  1  and the Trailing Constant is  3.
 The factor(s) are:
of the Leading Coefficient :Â Â 1
 of the Trailing Constant :  1 ,3
 Let us test
  P  Q  P/Q  F(P/Q)   Divisor     -1     1     -1.00     4.00        -3     1     -3.00     12.00        1     1     1.00     4.00        3     1     3.00     12.00  Â
Factoring:Â Â x2-6Â
Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)
Proof :  (A+B) • (A-B) =
        A2 - AB + BA - B2 =
        A2 - AB + AB - B2 =
         A2 - B2
Note :Â Â AB = BAÂ is the commutative property of multiplication.
Note :Â Â -Â ABÂ + ABÂ equals zero and is therefore eliminated from the expression.
Check : 6 is not a square !!(x2 + 3) • (x2 - 6) = 0 product of several terms equals zero.Â
 When a product of two or more terms equals zero, then at least one of the terms must be zero.Â
 We shall now solve each term = 0 separatelyÂ
 In other words, we are going to solve as many equations as there are terms in the productÂ
 Any solution of term = 0 solves product = 0 as well.Solve  :    x2+3 = 0Â
 Subtract  3 from both sides of the equation :Â
                      x2 = -3
Â
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get: Â
                      x =  ± √ -3 Â
 In Math,  i  is called the imaginary unit. It satisfies   i2  =-1. Both   i  and   -i  are the square roots of   -1Â
Accordingly,  √ -3  =
                    √ -1• 3   =
                    √ -1 •√  3   =
                    i •  √ 3
The equation has no real solutions. It has 2 imaginary, or complex solutions.
                      x= 0.0000 + 1.7321 iÂ
                      x= 0.0000 - 1.7321 iÂ
Solve  :    x2-6 = 0Â
 Add  6 to both sides of the equation :Â
                      x2 = 6
Â
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get: Â
                      x =  ± √ 6 Â
 The equation has two real solutions Â
 These solutions are  x = ± √6 = ± 2.4495 Â
 Solving  x4-3x2-18 = 0 directly
Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula
Solve   x4-3x2-18 = 0
This equation is reducible to quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using  w , such that  w = x2 transforms the equation into :
 w2-3w-18 = 0
Solving this new equation using the quadratic formula we get two real solutions :
  6.0000  or  -3.0000
Now that we know the value(s) of  w , we can calculate  x since  x  is  √ w Â
Doing just this we discover that the solutions of
   x4-3x2-18 = 0
  are either :Â
  x =√ 6.000 = 2.44949  or :
  x =√ 6.000 = -2.44949  or :
  x =√-3.000 = 0.0 + 1.73205 i  or :
  x =√-3.000 = 0.0 - 1.73205 i
so we found four answers:
 x = ± √6 = ± 2.4495
  x= 0.0000 - 1.7321 iÂ
  x= 0.0000 + 1.7321 iÂ
sorry for taking time to answer
hope its helps