Asst 12

course Mth 158

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3. Factor 6x^3 - 2x^2 2x^2(3x-1) 6x^3-2x^2 = 2x^2(3x-1) 6. True - A radical equation sometimes has no solution. Find the real solutions of each equation. 9. The square root of 3t+4 = -6 This has no real solution because a square root can't be negative. 12. No real solution because cannot have a negative in a square root. 15. 5 is in the check mark part of the square root and inside the square root is x^2 + 2x = -1 (5 is in the check mark part of the square root and then inside the square root is x^4 + 2x)^5 = (-1)^5 x^4 + 2x = -1 x^4 + 2x + 1 = 0 (x^2 + 1)(x^2 + 1) = 0 x^2 + 1 = 0 x^2 = -1 No real solution 18. x = 3 outside the square root of x (x)^2 = (3 and then the square root of x)^2 x^2 = 9x x^2 - 9x = 0 x(x-9) = 0 Solution set is {0, 9} 21. x = 2 and then the square root of x-1 x^2 = (2 and then the square root of x - 1)^2 x^2 = 4(x-1) x^2 = 4x - 4 x^2 - 4x + 4 = 0 (x-2)^2 = 0 x = 2 The solution set is {2} 25. 3 + the square root of 3x+1 = x The square root of 3x+1 = x-3 (The square root of 3x+1)^2 = (x-3)^2 3x+1 = x^2 - 6x +9 0 = x^2 - 9x + 8 (x-1)(x-8) = 0 Solution set is {1, 8} 28. The square root of 3x + 7 + the square root of x + 2 = 1 The square root of 3x + 7 = 1 + the square root of x + 2 (the square root of 3x + 7)^2 = (1 + the square root of x+2)^2 3x + 7 = 1=1 and the square root of x + 2 + x + 1 ?????I don't know how to do the rest of this problem??? 31. ?? 34. (3x - 5)^1/2 = (2)^2 ((3x-5)^1/2)^2 = (2)^2 3x - 5 = 4 3x = 3 The solution set is {3} 37. (x^2 + 9)^1/2 = 5 ((x^2 + 9)^1/2)^2 = (5)^2 x^2 + 9 = 25 x^2 = 16 x^2 = 16 The solution set is {-4, 4} 40. x^3/1 - 9x^1/4 = 0 x^3/4 = 9x^1/4 (x^3/4)^2 = (9x^1/4)^2 x^3 = 81x X3 - 81x = 0 x(x^2-81) = 0 x(x+9)(x-9) = 0 x = 0 or x = 9 Solution is {0, 9}

Very good strategy. However you should have used the 4th power. Note also the notation:
Assuming your second step
x^(3/4) = 9x^(1/4) take the 4th power of both sides: (x^(3/4))^4 = ( 9 x^(1/4) )^4 so x^(3/4 * 4) = 9^4 * x^(1/4 * 4) or x^3 = 9^4 * x. Thus x^3 - 9^4 * x = 0 and x ( x^2 - 9^4) = 0. Since sqrt(9^4) is 9^2 or 81 you have x ( x-81 ) ( x+81) = 0 so x = 0 or x = 81 or x = -81.
Solution set is {0, -81, 81}.

43. 3x^4 - 2x^2 - 1 = 0 (3x^2 + 1)(x^2 - 1) = 0 3x^2 + 1 = 0 3x^2 = -1 No real solution

Right so far, but
x^2 - 1 = 0
does have real solutions -1 and 1.

46. x^6 - 7x^3 - 8 = 0 (x^3+8)(x^3-1) = 0 x^3 + 8 = 0 x^3 = -8 x = -2 Solution is {-2}

Also x = 1. If x^3 - 1 = 0 the equation is also solved.

49. ?? 52. ?? 55. x + square root of x = 20 p = square root of x p^2 = x p^2 + p = 20 p^2 + p - 20 = 0 (p+5)(p-4) = 0 p = 5 or p = 4 p = 5 square root of x = 25 p = 4 square root of x 16 58. z^1/2 - 4z^1/4 + 4 = 0 p = z^1/2 p^2 = z^1/2 p^2 - 4p + 4 = 0 (p-4)(p+4)= 0 {p = 4, p = -4} 61. 4 is above the check mark in the square root problem and inside is 5x^2 - 6 = x (4 is above the check mark and 5x^2 - 6)^4 = x^4 5x^2 - 6 = x^4 0 = x^4 - 5x^2 + 6 p = x^2 p ^2 = x^4 0 = p^2 - 5p +6 (p-3)(p-2) = 0 p = 3 p = 2 64. ?? 66. ?? 69. 2x^2/3 - 5x^1/3 - 3 = 0 p = x^1/3 p^2 = x^2/3

right

p = -1/2

There is no need for this

2p^2 - 5p--3 = 0 (2p+1)(p-3) = 0 p = ^-1/2 or p = 3

the solution to 2p+1 = 0 is p = -1/2, not p = ^-1/2; this latter expression is in fact meaningless, and this might be what's confusing you.

p = -1/2 x^1/3 = ^-1/2

You mean x^(1/3) = -1/2. Raise both sides to the power 3 and you get x = -1/8.

(x^1/3)^3 = (-1/2)^3 I don't know how to finish this

The other solution is x^(1/3) = 3, so that x = 3^3 = 27.

72. (y/y-1)^2 = 6(y/y-1) + 7 y^2 = 6y + 7

I don't think your grouping is correct. Using what I think is the correct grouping you would let u = y / (y-1) and get u^2 = 6 u + 7, which rearranges to u^2 - 6 u - 7, etc.

75. 4x^3 = 3x^2 4x^3 - 3x^2 = 0 x^2(4x-3) = 0 x^2 = 0 x = 0 4x - 3 = 0 x = 3/4 solution is (0, 3/4) 78. x^3 + 6x^2 - 7x = 0 x^3 - 6x^2 + 7x = 0 x^3(6x^2 - 7) = 0

The third line is not correct.
x^3 - 6x^2 + 7x = 0 factors into x ( x^2 - 6x + 7) = 0.

81. x^3 - 3x^2 - 4x + 12 = 0 x^2(x-3)-4(x-3) = 0 (x-3)(x^2-4) = 0 (x-3)(x+2)(x-2) = 0 x-3 = 0 x = 3 x = -2 (-2, 2, 3)

Very good.

84. ?? 87. ?? 90. ?? "