Question

Answer both questions with work and process understandably shown.

Answer 1

15. x = -8

16. x = 1¼ (1.25)

Explanation:

15.

2x² - 5x - 3

2x² - 6x + x - 3

2x(x - 3) + 1(x - 3)

(2x + 1)(x - 3)

(x² - 23)/[(2x + 1)(x - 3)] + 2/(x - 3) = -1/(2x + 1)

Multiply the entire equation by (2x + 1)(x - 3)

x² - 23 + 2(2x + 1) = -1(x - 3)

x² - 23 + 4x + 2 + x - 3 = 0

x² + 5x - 24 = 0

x² + 8x - 3x - 24 = 0

x(x + 8) - 3(x + 8) = 0

(x - 3)(x + 8) = 0

x = -8, 3

x can not be 3 because the denominator becomes 0 at x = 3

So the only solution is x = -8

16.

3x² - x - 2

3x² - 3x + 2x - 2

3x(x - 1) + 2(x - 1)

(3x + 2)(x - 1)

(4x² - 24x)/[(3x + 2)(x - 1)] + 3/(3x + 2) = -4/(x - 1)

Multipy the entire equation by (3x + 2)(x - 1)

4x² - 24x + 3(x - 1) = -4(3x + 2)

4x² - 24x + 3x - 3 + 12x + 8 = 0

4x² - 9x + 5 = 0

4x² - 4x - 5x + 5 = 0

4x(x - 1) - 5(x - 1) = 0

(4x - 5)(x - 1) = 0

x = 5/4, 1

x can not be 1 because the denominator becomes 0 at x = 1

So the only solution is x = 5/4 or 1¼

Answer 2

see below

Explanation:

(x^2 -23) 2 -1

-------------------- + ------- = ------------

2x^2 -5x -3 x-3 2x+1

Factor the first term

(x^2 -23) 2 -1

-------------------- + ------- = ------------

(2x+1) (x-3) x-3 2x+1

Multiply both sides by (2x+1) (x-3) to get rid of the fractions

(x^2 -23) 2(2x+1) (x-3) -1

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

(2x+1) (x-3) x-3 2x+1

Canceling like terms in the numerator and denominator, we are left with

(x^2 -23) + 2(2x+1) = -1(x-3)

Distribute

x^2 -23 +4x+2 = -x+3

Add x to each side

x^2 -23 +4x+x+2 = -x+x+3

x^2 -23 +5x+2 = 3

Subtract 3 from each side

x^2 -23 +5x+2-3 = 3-3

x^2 +5x -24 =0

Factor

(x+8)(x-3) =0

Using the zero product property

x+8 =0 x-3=0

x=-8 x=3

4x^2 -24x 3 -4

-------------------- + ------- = ------------

3x^2 -x -2 3x+2 x-1

Factor the first term

4x^2 -24x 3 -4

-------------------- + ------- = ------------

(3x+2) (x-1) 3x+2 x-1

Multiply both sides by (3x+2) (x-1) to get rid of the fractions

4x^2 -24x 3 -4

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

(3x+2) (x-1) 3x+2 x-1

Canceling like terms in the numerator and denominator, we are left with

4x^2 -24x + 3 (x-1) =(3x+2)( -4)

Distribute

4x^2 -24x +3x-3 = -12x-8

Add 12x to each side

4x^2 -24x +3x-3 = -12x+12x-8

4x^2 -24x+3x+12x-3 = -8

Add 8 to each side

4x^2 -24x+3x+12x-3+8 = -8+8

4x^2 -9x+5 =0

Factor

(x-1)(4x-5) =0

Using the zero product property

x-1=0 4x-5=0

x=1 4x=5

x=1 x=5/4