use these numbers to answer the problems

Question

Solve by completing the square 3x²+3x=21

Use the discriminant to find the number of unique real solution

Solve with the quadratic formula

Answer 1

3x² + 3x = 21

 

1)  To complete the square, first factor out the 3:

 

3(x² + x) = 21

x² + x = 7

 

Now add a number "m" to each side:

 

x² + x + m = 7 + m

 

Set m = 1/2 of the coefficient of the x term (1), squared = (1/2)²:

 

(x² + x + 1/4) = 7 + 1/4

(x + 1/2)² = 29/4

(x + 1/2) = √(29/4)

x + 1/2 = ±(1/2)√29

 

x = -1/2 ± (1/2)√29

 

2)  To find the discriminant, put the original equation into standard form, y = ax² + bx + c:

 

3x² + 3x = 21

3x² + 3x - 21 = 0

Divide both sides by 3 to simplify:

x² + x - 7 = 0

 

The discriminant is:

 

b² - 4ac

(1)² - (4)(1)(-7) = 29

 

Since the discriminant is positive, there are two real solutions. 

 

3)  The solutions can be found using the quadratic formula:

 

x = (-b/2a) ± (1/2a)√(b²-4ac)

 

x = -1/2 ± (1/2)√29