(a+1/a)^2
Give the formulae
Answers
Step by step solution :
Step 1 :
1
Simplify —
a
Equation at the end of step 1 :
1
(a + —) - 2 = 0
a
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Adding a fraction to a whole
Rewrite the whole as a fraction using a as the denominator :
a a • a
a = — = —————
1 a
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:
a • a + 1 a2 + 1
————————— = ——————
a a
Equation at the end of step 2 :
(a2 + 1)
———————— - 2 = 0
a
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using a as the denominator :
2 2 • a
2 = — = —————
1 a
Polynomial Roots Calculator :
3.2 Find roots (zeroes) of : F(a) = a2 + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of a for which F(a)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers a 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
Adding fractions that have a common denominator :
3.3 Adding up the two equivalent fractions
(a2+1) - (2 • a) a2 - 2a + 1
———————————————— = ———————————
a a
Trying to factor by splitting the middle term
3.4 Factoring a2 - 2a + 1
The first term is, a2 its coefficient is 1 .
The middle term is, -2a its coefficient is -2 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 1 • 1 = 1
Step-2 : Find two factors of 1 whose sum equals the coefficient of the middle term, which is -2 .
-1 + -1 = -2 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -1 and -1
a2 - 1a - 1a - 1
Step-4 : Add up the first 2 terms, pulling out like factors :
a • (a-1)
Add up the last 2 terms, pulling out common factors :
1 • (a-1)
Step-5 : Add up the four terms of step 4 :
(a-1) • (a-1)
Which is the desired factorization
Multiplying Exponential Expressions :
3.5 Multiply (a-1) by (a-1)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (a-1) and the exponents are :
1 , as (a-1) is the same number as (a-1)1
and 1 , as (a-1) is the same number as (a-1)1
The product is therefore, (a-1)(1+1) = (a-1)2
Equation at the end of step 3 :
(a - 1)2
———————— = 0
a
Step 4 :
When a fraction equals zero :
4.1 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:
(a-1)2
—————— • a = 0 • a
a
Now, on the left hand side, the a cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
(a-1)2 = 0