Algebra math help please?

Question

Hi there, I have an algebra question, how would you solve these problems? (If you only know how to do one, then still answer please!) :) 

1. √x^2=21-√x^2-9 (square root includes nine) 


2. √x + √x-7 (root includes seven) = 21/√x-7 (root includes seven) 


3. 1/1-x + 1/1+√x=1/1-√x

Answer 1

3) 1/1-x + 1/1+√x = 1/1-√x ---> (1)

We know that (a^2-b^2) = (a+b)(a-b)

(1-x) = (1+√x)(1-√x)

From (1)

1/(1+√x)(1-√x) + 1/1+√x = 1/1-√x

1+(1-√x)]/(1+√x)(1-√x) = 1/(1-√x)

(2-√x)/(1+√x) = 1

Multipleto eachside by (1+√x).

2-√x = 1+√x

Subtract 1 from each side.

1-√x = √x

Add √x to each side.

1 = 2√x

Multiple to each side by 2.

√x = 1/2

Squaring on each side.

x = 1/4

Solution x = 1/4.

Answer 2

2) √x + √(x-7) = 21/√(x-7)

Multiple to each side by √(x-7).

√x (√(x-7))+√(x-7)(√(x-7)) = 21√(x-7)/√(x-7)

x(x-7)+(x-7)^2 = 21

(x-7)(x+x-7) = 21

(x-7)(2x-7) = 21

2x^2-7x-14x+49 = 21

Subtract 21 from each side.

2x^2-21x+49-21 = 0

2x^2-21x+28 = 0

Compare it quadratic equation ax^2+bx+c = 0.

a = 2, b = -21, c = 28

x = [21±√(441-224)]/4

x = [21±√217]/4

x = [21±14.73]/4

x = 35.73/4 and x = 6.27/4

x = 8.93 and x =  1.56

 

 

Comments

The solution is x = 16.

Answer 3

2) √x + √(x-7) = 21/√(x-7)

Multiple to each side by √(x-7).

√x(√(x-7))+√(x-7)(√(x-7)) = 21√(x-7)/√(x-7)

√x(x-7)+(x-7) = 21

√(x^2-7x)+x-7 = 21

Add 7 to each side.

√(x^2-7x)+x = 28

Subtract x from each side.

√(x^2-7x) = 28-x

Squring on each side.

(x^2-7x) = (28-x)^2

x^2-7x = 784+x^2-56x

Cancell the x^2 term on each side.

-7x = 784-56x

Add 56x to each side.

49x = 784

Divide to each side by 49.

x = 784/49

x = 16

Solution x = 16.

Answer 4

1) √x^2 = 21-√(x^2-9)

Squring on each side.

x^2 = 441+x^2-9-42√(x^2-9)

x^2 = 432+x^2-42√(x^2-9)

Add 42√(x^2-9) to each side.

42√(x^2-9)+x^2 = x^2+432

Cancell the x^2 terms.

42√(x^2-9) = 432

Divide to each side by 42.

42√(x^2-9)/42 = 432/42

√(x^2-9) = 10.28

Squring agian to each side.

x^2-9 = 105.67

Add 9 to each side.

x^2 = 114.6

Apply squre root on each side.

x = ±10.7

Solution x = ±10.7.

 

 

Comments

The solution is x = 75/7.

Answer 5

• 1).

The equation is √(x²) = 21 - √(x² - 9).

Rewrite the equation as x = 21 - √(x² - 9)      (∵ √(x²) = 1)

Subtract 21 from each side.

x - 21= 21 - 21 - √(x² - 9)

x - 21= - √(x² - 9)

Squaring on each side.

(x - 21)²= (- √(x² - 9))²

Apply square of difference : (a - b)² = a² - 2ab + a²

x² - 42x + 441 = x² - 9

Cancel common terms.

- 42x + 441 = - 9

Subtract  441 from each side.

- 42x + 441 - 441= - 9 - 441

- 42x = - 450

Divide each side by negative 42.

- 42x/- 42 = - 450/- 42

cancel common terms.

x = 450/42

⇒ x = 75/7.

The solution is x = 75/7.