Question

find the square root of 4x^4-12x^2+9

Answer 1

root 3 by 2 is answer

Answer 2

4x4-12x2+9=0 

Two solutions were found :

                   x = ±√ 1.500 = ± 1.22474

Step by step solution :

Step  1  :

Equation at the end of step  1  :

((4 • (x4)) - (22•3x2)) + 9 = 0

Step  2  :

Equation at the end of step  2  :

(22x4 - (22•3x2)) + 9 = 0

Step  3  :

Trying to factor by splitting the middle term

 3.1     Factoring  4x4-12x2+9 

The first term is,  4x4  its coefficient is  4 .
The middle term is,  -12x2  its coefficient is  -12 .
The last term, "the constant", is  +9 

Step-1 : Multiply the coefficient of the first term by the constant   4 • 9 = 36 

Step-2 : Find two factors of  36  whose sum equals the coefficient of the middle term, which is   -12 .

     -36   +   -1   =   -37     -18   +   -2   =   -20     -12   +   -3   =   -15     -9   +   -4   =   -13     -6   +   -6   =   -12   That's it


Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -6  and  -6 
                     4x4 - 6x2 - 6x2 - 9

Step-4 : Add up the first 2 terms, pulling out like factors :
                    2x2 • (2x2-3)
              Add up the last 2 terms, pulling out common factors :
                    3 • (2x2-3)
Step-5 : Add up the four terms of step 4 :
                    (2x2-3)  •  (2x2-3)
             Which is the desired factorization

Trying to factor as a Difference of Squares :

 3.2      Factoring:  2x2-3 

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 :  AB = BA is the commutative property of multiplication. 

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  2  is not a square !! 

Ruling : Binomial can not be factored as the
difference of two perfect squares

Trying to factor as a Difference of Squares :

 3.3      Factoring:  2x2-3 

Check :  2  is not a square !! 

Ruling : Binomial can not be factored as the
difference of two perfect squares

Multiplying Exponential Expressions :

 3.4    Multiply  (2x2-3)  by  (2x2-3) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (2x2-3)  and the exponents are :
          1 , as  (2x2-3)  is the same number as  (2x2-3)1 
 and   1 , as  (2x2-3)  is the same number as  (2x2-3)1 
The product is therefore,  (2x2-3)(1+1) = (2x2-3)2 

Equation at the end of step  3  :

(2x2 - 3)2 = 0

Step  4  :

Solving a Single Variable Equation :

 4.1      Solve  :    (2x2-3)2 = 0 

  (2x2-3) 2 represents, in effect, a product of 2 terms which is equal to zero 

For the product to be zero, at least one of these terms must be zero. Since all these terms are equal to each other, it actually means :   2x2-3  = 0 

Add  3  to both sides of the equation : 
                      2x2 = 3 
Divide both sides of the equation by 2:
                     x2 = 3/2 = 1.500 
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 3/2  

 The equation has two real solutions  
 These solutions are  x = ±√ 1.500 = ± 1.22474  
 

Supplement : Solving Quadratic Equation Directly

Solving  4x4-12x2+9  = 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

Solving a Single Variable Equation :

Equations which are reducible to quadratic :

 5.1     Solve   4x4-12x2+9 = 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 :
 4w2-12w+9 = 0

Solving this new equation using the quadratic formula we get one solution :
   w = 1.50000010.5 

Two solutions were found :

                   x = ±√ 1.500 = ± 1.22474


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