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)-2*x^3-(-38)=0
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((3 • (0 - (x3))) - 2x3) - -38 = 0
Step 2 :
Equation at the end of step 2 :
( -3x3 - 2x3) - -38 = 0
Step 3 :
Trying to factor as a Difference of Cubes:
3.1 Factoring: 38-5x3
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 : 38 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) = -5x3+38
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 38 and the Trailing Constant is -5.
The factor(s) are:
of the Leading Coefficient : 1,2 ,19 ,38
of the Trailing Constant : 1 ,5
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 43.00 | ||||||
| -1 | 2 | -0.50 | 38.62 | ||||||
| -1 | 19 | -0.05 | 38.00 | ||||||
| -1 | 38 | -0.03 | 38.00 | ||||||
| -5 | 1 | -5.00 | 663.00 | ||||||
| -5 | 2 | -2.50 | 116.12 | ||||||
| -5 | 19 | -0.26 | 38.09 | ||||||
| -5 | 38 | -0.13 | 38.01 | ||||||
| 1 | 1 | 1.00 | 33.00 | ||||||
| 1 | 2 | 0.50 | 37.38 | ||||||
| 1 | 19 | 0.05 | 38.00 | ||||||
| 1 | 38 | 0.03 | 38.00 | ||||||
| 5 | 1 | 5.00 | -587.00 | ||||||
| 5 | 2 | 2.50 | -40.12 | ||||||
| 5 | 19 | 0.26 | 37.91 | ||||||
| 5 | 38 | 0.13 | 37.99 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 3 :
38 - 5x3 = 0
Step 4 :
Solving a Single Variable Equation :
4.1 Solve : -5x3+38 = 0
Subtract 38 from both sides of the equation :
-5x3 = -38
Multiply both sides of the equation by (-1) : 5x3 = 38
Divide both sides of the equation by 5:
x3 = 38/5 = 7.600
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:
x = ∛ 38/5
The equation has one real solution
This solution is x = ∛ 7.600 = 1.96610
One solution was found :
x = ∛ 7.600 = 1.96610How did we do?
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