What are the solutions of the equation x4-9x^2+8=0 use

x = 2 • ± √2 = ± 2.8284

x = 1

x = -1

Step-by-step explanation:

x4-9x2+8=0  

Four solutions were found :

x = 2 • ± √2 = ± 2.8284

x = 1

x = -1

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "x2"   was replaced by   "x^2".  1 more similar replacement(s).

Step by step solution :

Step  1  :

Equation at the end of step  1  :

 ((x4) -  32x2) +  8  = 0  

Step  2  :

Trying to factor by splitting the middle term

2.1     Factoring  x4-9x2+8  

The first term is,  x4  its coefficient is  1 .

The middle term is,  -9x2  its coefficient is  -9 .

The last term, "the constant", is  +8  

Step-1 : Multiply the coefficient of the first term by the constant   1 • 8 = 8  

Step-2 : Find two factors of  8  whose sum equals the coefficient of the middle term, which is   -9 .

     -8    +    -1    =    -9    That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -8  and  -1  

                    x4 - 8x2 - 1x2 - 8

Step-4 : Add up the first 2 terms, pulling out like factors :

                   x2 • (x2-8)

             Add up the last 2 terms, pulling out common factors :

                    1 • (x2-8)

Step-5 : Add up the four terms of step 4 :

                   (x2-1)  •  (x2-8)

            Which is the desired factorization

Trying to factor as a Difference of Squares :

2.2      Factoring:  x2-1  

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 : 1 is the square of 1

Check :  x2  is the square of  x1  

Factorization is :       (x + 1)  •  (x - 1)  

Trying to factor as a Difference of Squares :

2.3      Factoring:  x2 - 8  

Check : 8 is not a square !!  

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

Equation at the end of step  2  :

 (x + 1) • (x - 1) • (x2 - 8)  = 0  

Step  3  :

Theory - Roots of a product :

3.1    A product of several terms equals zero.  

When a product of two or more terms equals zero, then at least one of the terms must be zero.  

We shall now solve each term = 0 separately  

In other words, we are going to solve as many equations as there are terms in the product  

Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

3.2      Solve  :    x+1 = 0  

Subtract  1  from both sides of the equation :  

                     x = -1  

Solving a Single Variable Equation :

3.3      Solve  :    x-1 = 0  

Add  1  to both sides of the equation :  

                     x = 1  

Solving a Single Variable Equation :

3.4      Solve  :    x2-8 = 0  

Add  8  to both sides of the equation :  

                     x2 = 8  

When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  

                     x  =  ± √ 8  

Can  √ 8 be simplified ?

Yes!   The prime factorization of  8   is

  2•2•2  

To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 8   =  √ 2•2•2   =

               ±  2 • √ 2  

The equation has two real solutions  

These solutions are  x = 2 • ± √2 = ± 2.8284  

Supplement : Solving Quadratic Equation Directly

Solving    x4-9x2+8  = 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 :

4.1     Solve   x4-9x2+8 = 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 :

w2-9w+8 = 0

Solving this new equation using the quadratic formula we get two real solutions :

  8.0000  or   1.0000

Now that we know the value(s) of  w , we can calculate  x  since  x  is  √ w  

Doing just this we discover that the solutions of  

  x4-9x2+8 = 0

 are either :  

 x =√ 8.000 = 2.82843  or :

 x =√ 8.000 = -2.82843  or :

 x =√ 1.000 = 1.00000  or :

 x =√ 1.000 = -1.00000  

Four solutions were found :

x = 2 • ± √2 = ± 2.8284

x = 1

x = -1

24/56=hx^7

step-by-step explanation: