Question

How do I solve for x in this problem X^3-4x^2-16x+64=0 ?

Answer 1

Two solutions were found :
1. x=4
2. x= -4

Step by step solution: Step 1 :

Equation at the end of step 1 :

(((x3) + 22x2) - 16x) - 64 = 0

Equation at the end of step 1 :

(((x3) + 22x2) - 16x) - 64 = 0 :

Step 2 :

Checking for a perfect cube :

2.1 x3+4x2-16x-64 is not a perfect cube

Trying to factor by pulling out :

2.2 Factoring: x3+4x2-16x-64

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: x3+4x2
Group 2: -16x-64

Pull out from each group separately :

Group 1: (x+4) • (x2)
Group 2: (x+4) • (-16)
-------------------
Add up the two groups :
(x+4) • (x2-16)
Which is the desired factorization


Trying to factor as a Difference of Squares :

2.3 Factoring: x2-16

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 : 16 is the square of 4
Check : x2 is the square of x1

Factorization is : (x + 4) • (x - 4)

Multiplying Exponential Expressions :

2.4 Multiply (x + 4) by (x + 4)

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

In our case, the common base is (x+4) and the exponents are :
1 , as (x+4) is the same number as (x+4)1
and 1 , as (x+4) is the same number as (x+4)1
The product is therefore, (x+4)(1+1) = (x+4)2

Equation at the end of step 2 :

(x + 4)2 • (x - 4) = 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+4)2 = 0

(x+4) 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 : x+4 = 0

Subtract 4 from both sides of the equation :
x = -4

Solving a Single Variable Equation :

3.3 Solve : x-4 = 0

Add 4 to both sides of the equation :
x = 4

Two solutions were found :

x = 4
x = -4

(Sorry the question is too long, but I hope you understand the question)