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