Solution - Linear equations with one unknown
Other Ways to Solve
Linear equations with one unknownStep by Step Solution
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
3*x^3-3-(3/3)=0
Step by step solution :
Step 1 :
1
Simplify —
1
Equation at the end of step 1 :
((3 • (x3)) - 3) - 1 = 0Step 2 :
Equation at the end of step 2 :
(3x3 - 3) - 1 = 0
Step 3 :
Trying to factor as a Difference of Cubes:
3.1 Factoring: 3x3-4
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0-b3 =
a3-b3
Check : 3 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Polynomial Roots Calculator :
3.2 Find roots (zeroes) of : F(x) = 3x3-4
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 3 and the Trailing Constant is -4.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,2 ,4
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -7.00 | ||||||
| -1 | 3 | -0.33 | -4.11 | ||||||
| -2 | 1 | -2.00 | -28.00 | ||||||
| -2 | 3 | -0.67 | -4.89 | ||||||
| -4 | 1 | -4.00 | -196.00 | ||||||
| -4 | 3 | -1.33 | -11.11 | ||||||
| 1 | 1 | 1.00 | -1.00 | ||||||
| 1 | 3 | 0.33 | -3.89 | ||||||
| 2 | 1 | 2.00 | 20.00 | ||||||
| 2 | 3 | 0.67 | -3.11 | ||||||
| 4 | 1 | 4.00 | 188.00 | ||||||
| 4 | 3 | 1.33 | 3.11 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 3 :
3x3 - 4 = 0
Step 4 :
Solving a Single Variable Equation :
4.1 Solve : 3x3-4 = 0
Add 4 to both sides of the equation :
3x3 = 4
Divide both sides of the equation by 3:
x3 = 4/3 = 1.333
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:
x = ∛ 4/3
The equation has one real solution
This solution is x = ∛ 1.333 = 1.10064
One solution was found :
x = ∛ 1.333 = 1.10064How did we do?
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