Solution - Quadratic equations

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "4.9" was replaced by "(49/10)".

Step by step solution :

Step  1  :

            49
 Simplify   ——
            10

Equation at the end of step  1  :

    49                  
  ((—— • x2) +  3x) +  10  = 0 
    10                  

Step  2  :

Equation at the end of step  2  :

   49x2           
  (———— +  3x) +  10  = 0 
    10            

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  10  as the denominator :

          3x     3x • 10
    3x =  ——  =  ———————
          1        10   

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:

 49x2 + 3x • 10     49x2 + 30x
 ——————————————  =  ——————————
       10               10    

Equation at the end of step  3  :

  (49x2 + 30x)    
  ———————————— +  10  = 0 
       10         

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  10  as the denominator :

          10     10 • 10
    10 =  ——  =  ———————
          1        10   

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   49x² + 30x  =   x • (49x + 30) 

Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions

 x • (49x+30) + 10 • 10     49x2 + 30x + 100
 ——————————————————————  =  ————————————————
           10                      10       

Trying to factor by splitting the middle term

 5.3     Factoring  49x² + 30x + 100 

The first term is,  49x²  its coefficient is  49 .
The middle term is,  +30x  its coefficient is  30 .
The last term, "the constant", is  +100 

Step-1 : Multiply the coefficient of the first term by the constant   49 • 100 = 4900 

Step-2 : Find two factors of  4900  whose sum equals the coefficient of the middle term, which is   30 .

For tidiness, printing of 48 lines which failed to find two such factors, was suppressed

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  5  :

  49x2 + 30x + 100
  ————————————————  = 0 
         10       

Step  6  :

When a fraction equals zero :

 6.1    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:

  49x2+30x+100
  ———————————— • 10 = 0 • 10
       10     

Now, on the left hand side, the  10  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   49x²+30x+100  = 0

Parabola, Finding the Vertex :

 6.2      Find the Vertex of   y = 49x²+30x+100

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 49 , is positive (greater than zero). 

 Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. 

 Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. 

 For any parabola,Ax²+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is  -0.3061  

 Plugging into the parabola formula  -0.3061  for  x  we can calculate the  y -coordinate : 
  y = 49.0 * -0.31 * -0.31 + 30.0 * -0.31 + 100.0
or   y = 95.408

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 49x²+30x+100
Axis of Symmetry (dashed)  {x}={-0.31} 
Vertex at  {x,y} = {-0.31,95.41} 
Function has no real roots

Solve Quadratic Equation by Completing The Square

 6.3     Solving   49x²+30x+100 = 0 by Completing The Square .

 Divide both sides of the equation by  49  to have 1 as the coefficient of the first term :
   x²+(30/49)x+(100/49) = 0

Subtract  100/49  from both side of the equation :
   x²+(30/49)x = -100/49

Now the clever bit: Take the coefficient of  x , which is  30/49 , divide by two, giving  15/49 , and finally square it giving  225/2401 

Add  225/2401  to both sides of the equation :
  On the right hand side we have :
   -100/49  +  225/2401   The common denominator of the two fractions is  2401   Adding  (-4900/2401)+(225/2401)  gives  -4675/2401 
  So adding to both sides we finally get :
   x²+(30/49)x+(225/2401) = -4675/2401

Adding  225/2401  has completed the left hand side into a perfect square :
   x²+(30/49)x+(225/2401)  =
   (x+(15/49)) • (x+(15/49))  =
  (x+(15/49))²
Things which are equal to the same thing are also equal to one another. Since
   x²+(30/49)x+(225/2401) = -4675/2401 and
   x²+(30/49)x+(225/2401) = (x+(15/49))²
then, according to the law of transitivity,
   (x+(15/49))² = -4675/2401

We'll refer to this Equation as  Eq. #6.3.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x+(15/49))²   is
   (x+(15/49))^2/2 =
  (x+(15/49))¹ =
   x+(15/49)

Now, applying the Square Root Principle to  Eq. #6.3.1  we get:
   x+(15/49) = √ -4675/2401

Subtract  15/49  from both sides to obtain:
   x = -15/49 + √ -4675/2401
In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 


Since a square root has two values, one positive and the other negative
   x² + (30/49)x + (100/49) = 0
   has two solutions:
  x = -15/49 + √ 4675/2401 •  i 
   or
  x = -15/49 - √ 4675/2401 •  i 

Note that  √ 4675/2401 can be written as
  √ 4675  / √ 2401   which is √ 4675  / 49

Solve Quadratic Equation using the Quadratic Formula

 6.4     Solving    49x²+30x+100 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  x  , the solution for   Ax²+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  x =   ————————
                      2A

  In our case,  A   =     49
                      B   =    30
                      C   =  100

Accordingly,  B²  -  4AC   =
                     900 - 19600 =
                     -18700

Applying the quadratic formula :

               -30 ± √ -18700
   x  =    ————————
                        98

In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written  (a+b*i) 

Both   i   and   -i   are the square roots of minus 1

Accordingly,√ -18700  = 
                    √ 18700 • (-1)  =
                    √ 18700  • √ -1   =
                    ±  √ 18700  • i

Can  √ 18700 be simplified ?

Yes!   The prime factorization of  18700   is
   2•2•5•5•11•17 
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).

√ 18700   =  √ 2•2•5•5•11•17   =2•5•√ 187   =
                ±  10 • √ 187

  √ 187   , rounded to 4 decimal digits, is  13.6748
 So now we are looking at:
           x  =  ( -30 ± 10 •  13.675 i ) / 98

Two imaginary solutions :

 x =(-30+√-18700)/98=(-15+5i√ 187 )/49= -0.3061+1.3954i
  or: 
 x =(-30-√-18700)/98=(-15-5i√ 187 )/49= -0.3061-1.3954i

Two solutions were found :

1.  x =(-30-√-18700)/98=(-15-5i√ 187 )/49= -0.3061-1.3954i
2.  x =(-30+√-18700)/98=(-15+5i√ 187 )/49= -0.3061+1.3954i