1. x^2-5x-14=0
2. 5x^2+12x=9
3. 6x^2=5x+6
: (

1. x = 7 or x = -2

2. x = 3/5 or x = -3

3. x = 3/2 or x = -2/3

Step-by-step explanation:

1. Solve for x over the real numbers:

x^2 - 5 x - 14 = 0

The left hand side factors into a product with two terms:

(x - 7) (x + 2) = 0

Split into two equations:

x - 7 = 0 or x + 2 = 0

Add 7 to both sides:

x = 7 or x + 2 = 0

Subtract 2 from both sides:

 x = 7 or x = -2

2> Solve for x over the real numbers:

5 x^2 + 12 x = 9

Divide both sides by 5:

x^2 + (12 x)/5 = 9/5

Add 36/25 to both sides:

x^2 + (12 x)/5 + 36/25 = 81/25

Write the left hand side as a square:

(x + 6/5)^2 = 81/25

Take the square root of both sides:

x + 6/5 = 9/5 or x + 6/5 = -9/5

Subtract 6/5 from both sides:

x = 3/5 or x + 6/5 = -9/5

Subtract 6/5 from both sides:

x = 3/5 or x = -3

3. Solve for x over the real numbers:

6 x^2 = 5 x + 6

Subtract 5 x + 6 from both sides:

6 x^2 - 5 x - 6 = 0

The left hand side factors into a product with two terms:

(2 x - 3) (3 x + 2) = 0

Split into two equations:

2 x - 3 = 0 or 3 x + 2 = 0

Add 3 to both sides:

2 x = 3 or 3 x + 2 = 0

Divide both sides by 2:

x = 3/2 or 3 x + 2 = 0

Subtract 2 from both sides:

x = 3/2 or 3 x = -2

Divide both sides by 3:

x = 3/2 or x = -2/3

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50