Solution - Nonlinear equations
Step by Step Solution
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
-x^2+18-(93)=0
Step by step solution :
Step 1 :
Step 2 :
Pulling out like terms :
2.1 Pull out like factors :
-x2 - 75 = -1 • (x2 + 75)
Polynomial Roots Calculator :
2.2 Find roots (zeroes) of : F(x) = x2 + 75
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 1 and the Trailing Constant is 75.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,5 ,15 ,25 ,75
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 76.00 | ||||||
| -3 | 1 | -3.00 | 84.00 | ||||||
| -5 | 1 | -5.00 | 100.00 | ||||||
| -15 | 1 | -15.00 | 300.00 | ||||||
| -25 | 1 | -25.00 | 700.00 | ||||||
| -75 | 1 | -75.00 | 5700.00 | ||||||
| 1 | 1 | 1.00 | 76.00 | ||||||
| 3 | 1 | 3.00 | 84.00 | ||||||
| 5 | 1 | 5.00 | 100.00 | ||||||
| 15 | 1 | 15.00 | 300.00 | ||||||
| 25 | 1 | 25.00 | 700.00 | ||||||
| 75 | 1 | 75.00 | 5700.00 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 2 :
-x2 - 75 = 0
Step 3 :
Solving a Single Variable Equation :
3.1 Solve : -x2-75 = 0
Add 75 to both sides of the equation :
-x2 = 75
Multiply both sides of the equation by (-1) : x2 = -75
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ -75
In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1
Accordingly, √ -75 =
√ -1• 75 =
√ -1 •√ 75 =
i • √ 75
Can √ 75 be simplified ?
Yes! The prime factorization of 75 is
3•5•5
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).
√ 75 = √ 3•5•5 =
± 5 • √ 3
The equation has no real solutions. It has 2 imaginary, or complex solutions.
x= 0.0000 + 8.6603 i
x= 0.0000 - 8.6603 i
Two solutions were found :
- x= 0.0000 - 8.6603 i
- x= 0.0000 + 8.6603 i
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