Q:

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

Accepted Solution

A:
Answer: x = 2 • ± √2 = ± 2.8284 x = 1 x = -1Step-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