help please :)

Question

Solve for x; be sure to check for extraneous answers (PLEASE SOLVE ALL):

 

A) x^2 -9=0

 

B) (x+1)^2 -16=0

 

C) 3√(x+2)= 12

 

D) √(x^2 -5x +2) =0

 

E) √(x-3) = √(4x-15)

 

F) √(3x-1) +3x=1

 

G) √(4x+1) - √(3x-2) =1

Answer 1

A)

 

x² - 9 = 0

 

 

This is a difference in perfect squares.

 

(x + 3)(x - 3) = 0

 

 

x = -3          and        x = 3

 

 

 

B)

 

(x + 1)²  - 16 = 0

 

 

Add 16 on both sides of the equation.

 

(x + 1)² = 16

 

 

Square-root both sides of the equation to undo the power.

 

x + 1 = ±√(16)

 

x + 1 = ±4

 

 

Subtract 1 on both sides of the equation.

 

x = -1 ± 4

 

 

Due to the plus/minus sign, there will be two solutions.

 

x = -1 - 4              and           x = -1 + 4

 

x = -5                   and           x = 3

 

 

C)

 

3√(x + 2) = 12

 

 

Square both sides of the equation to undo the square-root.

 

9(x + 2) = 144

 

 

Divide both sides of the equation by 9.

 

x + 2 = 16

 

 

Subtract  on both sides of the equation.

 

x = 14

 

 

D)

 

For this one, we use can either use the fact that square-root of zero is zero, or we can just square both sides of the equation.

 

x² - 5x + 2 = 0

 

 

Use the quadratic formula:

 

x = (-b ± √(b² - 4ac)) / 2a

 

where:

a = 1

b = -5

c = 2

 

Plug in these values into the formula to get your solutions.  You will get two solutions due to the plus/minus sign.

 

 

E)

 

Note that both sides of the equation has a term that is under the square-root.  We just equate the terms that are under the square-root.

 

x - 3 = 4x - 15

 

 

Subtract x on both sides of the equation.

 

-3 = 3x - 15

 

 

Add 15 on both sides of the equation.

 

12 = -3x

 

 

Divide both sides of the equation by -3.

 

-4 = x

 

 

F)

 

Subtract 3x on both sides of the equation.

 

√(3x - 1) = -3x + 1

 

 

Square both sides of the equation to undo the square-root.

 

3x - 1 = (-3x + 1)(-3x + 1)

 

3x - 1 = 9x² - 6x + 1

 

 

Move all the terms to the right side of the equation to make the left side equal to zero.

 

0 = 9x² - 9x + 2

 

 

Factor using FOIL, if possible.  Otherwise, use quadratic formula.

 

0 = (3x - 1)(3x - 2)

 

 

Set the factors equal to zero.

 

3x - 1 = 0             and              3x - 2 = 0

 

 x = 1/3                and               x = 2/3

 

 

G)

 

Square both sides of the equation.

 

(√(4x + 1) - √(3x - 2))(√(4x + 1) - √(3x - 2)) = 1

 

 

Expand the left side using FOIL.

 

4x + 1 - 2√(12x² - 5x - 2) + (3x - 2) = 1

 

4x + 1 - 2√(12x² - 5x - 2) + 3x - 2 = 1

 

 

Combine like terms on the left side.

 

7x - 1 - 2√(12x² - 5x - 2) = 1

 

 

Move all the non-square-root terms to the right side of equation.

 

-2√(12x² - 5x - 2) = -7x + 2

 

 

Square both sides of the equation to undo the last square-root.

 

4(12x² - 5x - 2) = (-7x + 2)(-7x + 2) =

 

48x² - 20x - 8 = 49x² - 28x + 4

 

 

Move all the terms to the right side of equation to make the left side equal to zero.

 

0 = x² - 8x + 12

 

 

Factor using FOIL, if possible.

 

0 = (x - 6)(x - 2)

 

x = 6          and            x = 2

 

 

 

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