I need some help with these math problems.?

Question

1. What are x-intercepts to the nearest hundredth of y=-33x² + 68x +13?

2. Factor and simplify the following: 16(3x+1)² - 25(y+3)²

3. Determine the nature of the roots of the quadratic equation below. -2x² + 3x + 8 =0

4. Determine the value of "k" for which 3x² - 5x + k = 0

Answer 1

(1)

The Equation is y = -33x²+68x+13

To find the X-intercepts of the Equation, we make y = 0.

-33x²+68x+13 = 0

To find the roots of the equation, we use the formula where a= -33 , b = 68 and c = 13.

Roots are and

Roots are -0.17 and 2.23

Therefore x -intercepts points are (-0.17,0) and (2.23 , 0).

Answer 2

(2)

The Expression is 16(3x+1)² - 25(y+3)².

16(3x+1)² - 25(y+3)²

= 16(9x²+1+6x) - 25(y²+9+6y)

= 144x² + 16 + 96x - (25y² + 225 + 150y)

= 144x² + 16 + 96x - 25y² - 225 - 150y

= 144x² - 25y² + 96x - 150y - 209

Therefore the simplified Expression is 144x² - 25y² + 96x - 150y - 209.

Answer 3

(3)

The roots of the Quadratic equation is -2x² + 3x + 8 = 0.

To find the roots of the equation, we use the formula where a= -2 , b = 3 and c = 8.

Roots are and 

Roots are -1.386 and 2.886

Therefore the nature of roots are real and irrational in nature.

Answer 4

(4)

The Quadratic Expression is 3x² - 5x + k = 0.

To find the roots of the equation, we use the formula where a= 3 , b = -5 and c = k.

There are three cases,

Case 1) , then roots are real.

Case 2) , then roots are equal.

Case 3) , then roots are imaginary

Therefore the Value of the k depends on the nature of the root.