Solve each of the following quadratic equations by completing the square.?

Question

22. x2 - 6x - 3 = 0

28. 2x2 + 10x + 11 = 0

10. 2x2 - 5x = 3

16. 3x2 - 2x + 1 = 0

48. Construction. A garden area is 30 ft long and 20 ft wide. A path of uniform width
is set around the edge. If the remaining garden area is 400 ft2, what is the width of
the path?

Answer 1

x^2 - 6x + 9 = 12
(x - 3)^2 = 12
x - 3 = +/- sqrt(12)
x = 3 +/- sqrt(12)

2x^2 + 10x + 11
2x^2 + 10x = -11
x^2 + 5x = -5.5
x^2 + 5x + 6.25 = -5.5 + 6.25
(6.25 = (.5*5)^2 - half of the x coefficient squared)
(x + 2.5)^2 = .75
x + 2.5 = +/- sqrt(.75)
x = -2.5 +/- sqrt(.75)

2x^2 - 5x = 3
x^2 - 2.5x = 1.5
x^2 - 2.5 + 1.5625 = 1.5 + 1.5625
(x - 1.25)^2 = 3.0625
x - 1.25 = +/- sqrt(3.0625)
x = 1.25 +/- sqrt(3.0625)

3x^2 - 2x + 1 = 0
3x^2 - 2x = -1
x^2 - 2/3*x = -1/3
x^2 - 2/3*x + 1/9 = -1/3 + 1/9
(x - 1/3)^2 = -2/9
x - 1/3 = +/- sqrt(-2/9) this is irrational
x = 1/3 +/- sqrt(2/9)i

(30 - 2x) * (20 - 2x) = 400
600 - 100x + 4x^2 = 400
4x^2 - 100x = -200
x^2 - 25x = -50
x^2 - 25x + 156.25 = -50 + 156.25
(x - 12.5)^2 = 106.25
x - 12.5 = +/- sqrt(106.25)
x = 12.5 +/- sqrt(106.25)
x = 12.5 +/- 10.31
x = 2.19 or 25.81

The path can't be 25.81, or it would be bigger than the garden. It's 2.19 ft.

Answer 2

the way is as follows: a million) Take the coeffecient of x^2 as a typical element from the words containing x^2 & x ex. 2 (x^2 - 2x ) +a million 2) upload and subtract the sqaur of a million/2 the coeffecient of x i.e. 2( x^2 - 2x +a million -a million )+a million 3) write the complet sq. U get i.e. 2(x-a million)^2 -2 +a million = 0 2(x-a million)^2= a million (x -a million)^2 = a million/2 x-a million= +/- sqrt(2)/2 x = a million +/- sqrt(2)/2

Answer 3

22)
x² - 6x -3 = 0
(x - 3)² = x² - 6x + 9

So if you want to have (x-3)² you need to subtract 9, your equation is now:
(x-3)² - 9 - 3=0
(x-3)² - 12 = 0
(x-3)² = 12
x - 3 = ±√12 = ±2√3
x = 3±2√3

In general to complete the square:

x² - ax = (x - a/2)² - (a/2)², just watch out for signs

Answer 4

x^2 - 6x - 3 = 0
x^2 - 6x = 3
x^2 - 6x + 9 = 12
(x - 3)^2 = 12
x - 3 = sqrt(12)
x = 3 ± sqrt(12)

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2x^2 + 10x + 11 = 0
2x^2 + 10x = -11
x^2 + 5x = (-11/2)
x^2 + 5x + (25/4) = (3/4)
(x + (5/2))^2 = (3/4)
x + (5/2) = (1/2)sqrt(3)
x = (-5/2) ± (1/2)sqrt(3)
x = (1/2)(-5 ± sqrt(3))

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1.)
2x^2 - 5x = 3
x^2 - (5/2)x = (3/2)
x^2 - (5/2)x + (25/16) = (49/16)
(x - (5/4))^2 = (49/16)
x - (5/4) = ±(7/4)
x = (5/4) ± (7/4)
x = (-2/4) or (12/4)
x = (-1/2) or 3

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3x^2 - 2x + 1 = 0
3x^2 - 2x = -1
x^2 - (2/3)x = (-1/3)
x^2 - (2/3)x + (1/9) = (-2/9)
(x - (1/3))^2 = (-2/9)
x - (1/3) = ±(i/3)sqrt(2)
x = (1/3) ± (i/3)sqrt(2)
x = (1/3)(1 ± isqrt(2))