Solution - Simplifying radicals

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     (t^2/20)+8-(15)=0 

Step by step solution :

Step  1  :

            t2
 Simplify   ——
            20

Equation at the end of step  1  :

   t2          
  (—— +  8) -  15  = 0 
   20          

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  20  as the denominator :

         8     8 • 20
    8 =  —  =  ——————
         1       20  

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:

 t2 + 8 • 20     t2 + 160
 ———————————  =  ————————
     20             20   

Equation at the end of step  2  :

  (t2 + 160)    
  —————————— -  15  = 0 
      20        

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  20  as the denominator :

          15     15 • 20
    15 =  ——  =  ———————
          1        20   

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(t) = t² + 160
Polynomial Roots Calculator is a set of methods aimed at finding values of  t  for which   F(t)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  t  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  160.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,5 ,8 ,10 ,16 ,20 ,32 ,40 , etc

 Let us test ....

Note - For tidiness, printing of 15 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 3.3       Adding up the two equivalent fractions

 (t2+160) - (15 • 20)     t2 - 140
 ————————————————————  =  ————————
          20                 20   

Trying to factor as a Difference of Squares :

 3.4      Factoring:  t² - 140 

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 : 140 is not a square !!

Ruling : Binomial can not be factored as the difference of two perfect squares.

Equation at the end of step  3  :

  t2 - 140
  ————————  = 0 
     20   

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:

  t2-140
  —————— • 20 = 0 • 20
    20  

Now, on the left hand side, the  20  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   t²-140  = 0

Solving a Single Variable Equation :

 4.2      Solve  :    t²-140 = 0 

 Add  140  to both sides of the equation : 
                      t² = 140
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      t  =  ± √ 140  

 Can  √ 140 be simplified ?

Yes!   The prime factorization of  140   is
   2•2•5•7 
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).

√ 140   =  √ 2•2•5•7   =
                ±  2 • √ 35

The equation has two real solutions  
 These solutions are  t = 2 • ± √35 = ± 11.8322  
 

Two solutions were found :