Solution - Simplifying radicals
Step by Step Solution
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(2x2 + 12) + 36 = 0
Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
2x2 + 48 = 2 • (x2 + 24)
Polynomial Roots Calculator :
3.2 Find roots (zeroes) of : F(x) = x2 + 24
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 24.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,12 ,24
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 25.00 | ||||||
| -2 | 1 | -2.00 | 28.00 | ||||||
| -3 | 1 | -3.00 | 33.00 | ||||||
| -4 | 1 | -4.00 | 40.00 | ||||||
| -6 | 1 | -6.00 | 60.00 | ||||||
| -8 | 1 | -8.00 | 88.00 | ||||||
| -12 | 1 | -12.00 | 168.00 | ||||||
| -24 | 1 | -24.00 | 600.00 | ||||||
| 1 | 1 | 1.00 | 25.00 | ||||||
| 2 | 1 | 2.00 | 28.00 | ||||||
| 3 | 1 | 3.00 | 33.00 | ||||||
| 4 | 1 | 4.00 | 40.00 | ||||||
| 6 | 1 | 6.00 | 60.00 | ||||||
| 8 | 1 | 8.00 | 88.00 | ||||||
| 12 | 1 | 12.00 | 168.00 | ||||||
| 24 | 1 | 24.00 | 600.00 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 3 :
2 • (x2 + 24) = 0
Step 4 :
Equations which are never true :
4.1 Solve : 2 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Solving a Single Variable Equation :
4.2 Solve : x2+24 = 0
Subtract 24 from both sides of the equation :
x2 = -24
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ -24
In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1
Accordingly, √ -24 =
√ -1• 24 =
√ -1 •√ 24 =
i • √ 24
Can √ 24 be simplified ?
Yes! The prime factorization of 24 is
2•2•2•3
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).
√ 24 = √ 2•2•2•3 =
± 2 • √ 6
The equation has no real solutions. It has 2 imaginary, or complex solutions.
x= 0.0000 + 4.8990 i
x= 0.0000 - 4.8990 i
Two solutions were found :
- x= 0.0000 - 4.8990 i
- x= 0.0000 + 4.8990 i