the roots of the equation: 6(x²+1/x²)-25(x-1/x)+12=0
Answers
Answer:
6(x2+1/x2)-25(x-1/x)+12=0
Four solutions were found :
x = 3
x = 2
x = -1/2 = -0.500
x = -1/3 = -0.333
Step by step solution :
Step 1 :
1
Simplify —
x
Equation at the end of step 1 :
1 1
((6•((x2)+————))-(25•(x-—)))+12 = 0
(x2) x
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using x as the denominator :
x x • x
x = — = —————
1 x
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
2.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
x • x - (1) x2 - 1
——————————— = ——————
x x
Equation at the end of step 2 :
1 (x2-1)
((6•((x2)+————))-(25•——————))+12 = 0
(x2) x
Step 3 :
Trying to factor as a Difference of Squares :
3.1 Factoring: x2-1
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Step 4 :
1
Simplify ——
x2
Equation at the end of step 4 :
1 25•(x+1)•(x-1)
((6•((x2)+——))-——————————————)+12 = 0
x2 x
x2 x2 • x2
x2 = —— = ———————
1 x2
x2 • x2 + 1 x4 + 1
——————————— = ——————
x2 x2
Equation at the end of step 5 :
(x4+1) 25•(x+1)•(x-1)
((6•——————)-——————————————)+12 = 0
x2 x
Step 6 :
Polynomial Roots Calculator :
6.1 Find roots (zeroes) of : F(x) = x4+1
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 1.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 2.00
1 1 1.00 2.00
Polynomial Roots Calculator found no rational roots
Equation at the end of step 6 :
6•(x4+1) 25•(x+1)•(x-1)
(————————-——————————————)+12 = 0
x2 x
Step 7 :
Calculating the Least Common Multiple :
7.1 Find the Least Common Multiple
The left denominator is : x2
The right denominator is : x
Number of times each Algebraic Factor
appears in the factorization of:
Algebraic
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
x 2 1 2
Least Common Multiple:
x2
Calculating Multipliers :
7.2 Calculate multipliers for the two fractions
Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno
Left_M = L.C.M / L_Deno = 1
Right_M = L.C.M / R_Deno = x
Equation at the end of step 8 :
(3x + 1) • (2x + 1) • (x - 2) • (x - 3)
——————————————————————————————————————— = 0
x2
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