If x - 1/x = 15/4 then find the value of x + 1/x
Answers
Answer:
Step-by-step explanation:
Step-by-step explanation:
Two solutions were found :
x = -1/4 = -0.250
x = 4
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
x-(1/x)-(15/4)=0
Step by step solution :
Step 1 :
15
Simplify ——
4
Equation at the end of step 1 :
1 15
(x - —) - —— = 0
x 4
Step 2 :
1
Simplify —
x
Equation at the end of step 2 :
1 15
(x - —) - —— = 0
x 4
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 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 :
3.2 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 3 :
(x2 - 1) 15
———————— - —— = 0
x 4
Step 4 :
Trying to factor as a Difference of Squares :
4.1 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
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 1 is the square of 1
Check : x2 is the square of x1
Factorization is : (x + 1) • (x - 1)
Calculating the Least Common Multiple :
4.2 Find the Least Common Multiple
The left denominator is : x
The right denominator is : 4
Number of times each prime factor
appears in the factorization of:
Prime
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
2 0 2 2
Product of all
Prime Factors 1 4 4
Number of times each Algebraic Factor
appears in the factorization of:
Algebraic
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
x 1 0 1
Least Common Multiple:
4x
Calculating Multipliers :
4.3 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 = 4
Right_M = L.C.M / R_Deno = x
Making Equivalent Fractions :
4.4 Rewrite the two fractions into equivalent fractions
Two fractions are called equivalent if they have the same numeric value.
For example : 1/2 and 2/4 are equivalent, y/(y+1)2 and (y2+y)/(y+1)3 are equivalent as well.
To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.
L. Mult. • L. Num. (x+1) • (x-1) • 4
—————————————————— = —————————————————
L.C.M 4x
R. Mult. • R. Num. 15 • x
—————————————————— = ——————
L.C.M 4x
Adding fractions that have a common denominator :
4.5 Adding up the two equivalent fractions
(x+1) • (x-1) • 4 - (15 • x) 4x2 - 15x - 4
———————————————————————————— = —————————————
4x 4x
Trying to factor by splitting the middle term
4.6 Factoring 4x2 - 15x - 4
The first term is, 4x2 its coefficient is 4 .
The middle term is, -15x its coefficient is -15 .
The last term, "the constant", is -4
Step-1 : Multiply the coefficient of the first term by the constant 4 • -4 = -16
Step-2 : Find two factors of -16 whose sum equals the coefficient of the middle term, which is -15 .
-16 + 1 = -15 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -16 and 1
4x2 - 16x + 1x - 4
Step-4 : Add up the first 2 terms, pulling out like factors :
4x • (x-4)
Add up the last 2 terms, pulling out common factors :
1 • (x-4)
Step-5 : Add up the four terms of step 4 :
(4x+1) • (x-4)
Which is the desired factorization
Equation at the end of step 4 :
(x - 4) • (4x + 1)
—————————————————— = 0
4x
Step 5 :
When a fraction equals zero :
5.1 When a fraction equals zero ...
Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.
Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.
Here's how:
(x-4)•(4x+1)
———————————— • 4x = 0 • 4x
4x
Now, on the left hand side, the 4x cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
(x-4) • (4x+1) = 0
Theory - Roots of a product :
5.2 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