Problem 10

use the quadratic formula to solve each of the following quadratic equations.

2x^2-5x = 3

My answer:
solutions are 3 and - (1)/(2)

Problem 16

3x^2-2x+1=0

My answer:
(2+- radical (-8))/(6)

To solve a quadratic equation using the quadratic formula, follow these steps:

1. Write down the equation in the standard form: ax^2 + bx + c = 0.

Problem 10:
The given equation is: 2x^2 - 5x = 3
Rearrange the equation: 2x^2 - 5x - 3 = 0

Problem 16:
The given equation is: 3x^2 - 2x + 1 = 0

2. Identify the values of a, b, and c from the equation.

Problem 10:
a = 2, b = -5, c = -3

Problem 16:
a = 3, b = -2, c = 1

3. Plug in the values of a, b, and c into the quadratic formula:

[-b +- sqrt(b^2 - 4ac)]
x = --------------------------------------
2a

4. Simplify the equation inside the square root if needed.

Problem 10:
For this equation, b^2 - 4ac = (-5)^2 - 4(2)(-3) = 25 + 24 = 49

Problem 16:
For this equation, b^2 - 4ac = (-2)^2 - 4(3)(1) = 4 - 12 = -8

5. Apply the square root and simplify further if possible.

Problem 10:
[-(-5) +- sqrt(49)] / (2 * 2) = (5 +- 7) / 4

Problem 16:
[2 +- sqrt(-8)] / (2 * 3) = (2 +- 2sqrt(2)i) / 6

6. Finalize the solutions.

Problem 10:
(5 + 7) / 4 = 12 / 4 = 3
(5 - 7) / 4 = -2 / 4 = -1/2

Problem 16:
(2 + 2sqrt(2)i) / 6
(2 - 2sqrt(2)i) / 6

So the solutions for Problem 10 are 3 and -1/2, and for Problem 16, the solutions are (2 + 2sqrt(2)i)/6 and (2 - 2sqrt(2)i)/6.