solve by completing the square

Question

3x2to the second power+3x=21.                         Use the discriminant to find the number of unique real solution.                                              Solve with the quadratic formula

Answer 1

3x²  + 3x = 21

 

We want the left side of the equation to have a binomial squared term.  Lets divide both sides of equation by 3 to simplify the equation since  3x * x  is  3x²   and  

(3x + a)(x + a) is not a perfect square.

  

x² + x = 7

 

 

Next, is to find a binomial so that then it is squared and expanded, the first term is x² and the middle term is x.

 

(x + (1/2))² = (x + (1/2))(x + (1/2)) = x² + x + (1/4)

 

 

Add 1/4 on both sides of equation, so that we will have a polynomial on the left when factored will be a square.

 

x² + x + (1/4) = 7 + (1/4)

 

(x + (1/2))² = (28 + 1) / 4

(x + (1/2))²  = 29 / 4

 

 

Square-root both sides of equation.

 

x + 1/2 = ±√(29 / 4)

x + 1/2 = ± √(29) / 2

 

x = -(1/2) ± √(29) / 2

 

x = (-1 ± √29) / 2

 

x = (-1 + √29) / 2            and         x = (-1 - √29) / 2

 

 

The discriminant in the solution is the square-root part.  The value under the square-root is 29, which is a positive number.  We will have two real solutions.