How to solve equations?

Question

Solve each equation by completing the square.

1) x^2-3x-88=0

2) 2x^2+11x-21=0

3) z^2-2z=24

Fin the discriminant of each equation and describe the nature of the roots of the equation. Then solve each equation by using the quadratic formula.

4) 4x^2+19x-5=0

5) 2w^2+3w+3=0

6) 6b^2-39b+45=0

 

Answer 1

Hi Giovanni,

 

Lots of problems !!

 

We'll consider each one in turn:

 

 

Problem #1:

 

x² - 3x - 88 = 0

x² - 3x = 88

x² - 3x + [(1/2)(-3)]² = 88 + [(1/2)(-3)]²

x² - 3x + 9/4 = 88 + 9/4

(x - 3/2)² = 352/4 + 9/4

(x - 3/2)² = 361/4

x - 3/2 = +/- √(361/4)

x - 3/2 = +/- (19/2)

x = 3/2 +/- (19/2)

x = (3+19)/2

x = 11

x = (3-19)/2

x = -8

 

 

Problem #2:

 

2x² + 11x - 21 = 0

x² + (11/2)x = 21/2

x² + (11/2)x + [(1/2)(11/2)]² = 21/2 + [(1/2)(11/2)]²

x² + (11/2)x + 121/16 = 21/2 + 121/16

(x + 11/4)² = 168/16 + 121/16

(x + 11/4)² = 289/16

x + 11/4 = √(289/16)

x + 11/4 = +/- 17/4

x = -11/4 +/- 17/4

x = (-11+17)/4

x = 3/2

x = (-11-17)/4

x = -7

 

 

Problem #3:

 

z² - 2z = 24

z² - 2z + [(1/2)(-2)]² = 24 + [(1/2)(-2)]²

z² - 2z + 1 = 24 + 1

(z - 1)² = 25

z - 1 = +/- 5 

z = 1 +/- 5

z = 1 + 5

z = 6

z = 1 - 5

z = -4

 

 

Problem #4:

 

4x² + 19x - 5 = 0

a = 4; b = 19; c = -5

discriminant:  b² - 4ac

b² - 4ac = (19)² - 4(4)(-5)

b² - 4ac = 361 + 80

b² - 4ac = 441 (roots are real and rational)

quadratic formula:  x = [-b +/- √(b² - 4ac)] / 2a

x = [-19 +/- √441] / 2(4)

x = (-19 +/- 21) / 8

x = (-19+21)/8

x = 1/4

x = (-19-21)/8

x = -5

 

 

Problem #5:

 

2w² + 3w + 3 = 0

a = 2; b = 3; c = 3
discriminant: b² - 4ac
b² - 4ac = (3)² - 4(2)(3)
b² - 4ac = 9 - 24
b² - 4ac = -15 (roots are imaginary and irrational)
quadratic formula: w = [-b +/- √(b² - 4ac)] / 2a
w = [-3 +/- √(-15)] / 2(2)
w = (-3 +/- √15i) / 4
w = (-3+√15i)/4

w = (-3-√15i)/4

 

 

Problem #6:

 

6b² - 39b + 45 = 0

a = 6; b = -39; c = 45
discriminant: b² - 4ac
b² - 4ac = (-39)² - 4(6)(45)
b² - 4ac = 1521 - 1080 
b² - 4ac = 441 (roots are real and rational)

quadratic formula: b = [-b +/- √(b² - 4ac)] / 2a
b = [-(-39) +/- √441] / 2(6)
b = (39 +/- 21) / 12
b = (39+21)/12
b = 5

b = (39-21)/12
b = 3/2

 

 

Thanks for submitting these problems & glad to help.

 

God bless, Jordan.