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# Use 4r^2-28r=49 to answer problems

1. Solve by completing square
2. Use the discriminant to find the number of unique real solutions

### 2 Answers by Expert Tutors

SURENDRA K. | An experienced,patient & hardworking tutorAn experienced,patient & hardworking tut...
1
4r^2  -  28r -49=0

4r^2 -28r + 49 -49 -49=0

4r^2 -28r + 49 = 98

(2r-7)^2 =  98

2r-7 = 9.899

2r = 16.899

r= 8.449

thanks so much the help is very appreciated!
Philip P. | Effective and Affordable Math TutorEffective and Affordable Math Tutor
5.0 5.0 (443 lesson ratings) (443)
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Since Surendra answered the first question, I'll answer the next two. Let's put the equation into standard form:

4r2 - 28r - 49 = 0

2.  The discriminant is b2 - 4ac  where a=4, b= -28, and c= -49.

(-28)2 - (4)(4)(-49) = 784 + 784 = 1568

Since the discriminant is greater than zero, there will be two real solutions.  (The rules are: discriminant<0, no real solutions; discriminant=0, one solution; discriminant>0, two solutions).

3. Solve using the quadratic formula:

r = (-b/2a) ± (1/2a)√(b2-4ac)
Where a=4, b=-28, c=-49.

r = -(-28)/8 ± (1/8)√1568
r = 7/2 ± (7/2)√2

r =(7/2)(1+√2),  (7/2)(1-√2)   These are the exact solutions.  The approximate solutions are (rounded to two decimal places): r ≅ 8.45, -1.45

CHECK:
4{(7/2)(1+√2)}2 - 28{(7/2)(1+√2)} - 49 = 0
49 + 98√2 + 98 - 98 - 98√2 -49 = 0
49 - 49 = 0
CHECK!

4{(7/2)(1-√2)}2 - 28{(7/2)(1-√2)} - 49 = 0
49 - 98√2 + 98 - 98 + 98√2 - 49 = 0
49 - 49 = 0
CHECK !

1. Here's another take on problem 1 (complete the square):
4r2 - 28r = 49
4(r2 - 7r) = 49
r2 - 7r = 49/4
r2 - 7r + (-7/2)2 = 49/4 + (-7/2)2
(r - (7/2))2 = 98/4
r - 7/2 = ±√(98/4) = ±(7/2)√2

r = (7/2) ± (7/2)√2= (7/2)(1±√2)