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