Solution - Nonlinear equations
Other Ways to Solve
Nonlinear equationsStep by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "d1" was replaced by "d^1".
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
4*d^2*d^1-(-3*d)=0
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(22d2 • d) - -3d = 0Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
4d3 + 3d = d • (4d2 + 3)
Polynomial Roots Calculator :
3.2 Find roots (zeroes) of : F(d) = 4d2 + 3
Polynomial Roots Calculator is a set of methods aimed at finding values of d for which F(d)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers d 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 4 and the Trailing Constant is 3.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,3
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 7.00 | ||||||
| -1 | 2 | -0.50 | 4.00 | ||||||
| -1 | 4 | -0.25 | 3.25 | ||||||
| -3 | 1 | -3.00 | 39.00 | ||||||
| -3 | 2 | -1.50 | 12.00 | ||||||
| -3 | 4 | -0.75 | 5.25 | ||||||
| 1 | 1 | 1.00 | 7.00 | ||||||
| 1 | 2 | 0.50 | 4.00 | ||||||
| 1 | 4 | 0.25 | 3.25 | ||||||
| 3 | 1 | 3.00 | 39.00 | ||||||
| 3 | 2 | 1.50 | 12.00 | ||||||
| 3 | 4 | 0.75 | 5.25 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 3 :
d • (4d2 + 3) = 0
Step 4 :
Theory - Roots of a product :
4.1 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 :
4.2 Solve : d = 0
Solution is d = 0
Solving a Single Variable Equation :
4.3 Solve : 4d2+3 = 0
Subtract 3 from both sides of the equation :
4d2 = -3
Divide both sides of the equation by 4:
d2 = -3/4 = -0.750
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
d = ± √ -3/4
In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1
Accordingly, √ -3/4 =
√ -1• 3/4 =
√ -1 •√ 3/4 =
i • √ 3/4
The equation has no real solutions. It has 2 imaginary, or complex solutions.
d= 0.0000 + 0.8660 i
d= 0.0000 - 0.8660 i
Three solutions were found :
- d= 0.0000 - 0.8660 i
- d= 0.0000 + 0.8660 i
- d = 0
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