# Quadratic Equations

### Review ^2 means to the power of 2

## Expanding

(x + a) (x + b) = x^2 + bx + ax + ab

= x^2 + ax + bx + ab

= x^2 + (a + b) x + ab

Example:

(x + 2) (x + 6)

= x^2 + 5x + 2x + 10

= x^2 + 7x + 10

Expand the following

(x - 4) (x + 7)

(x - 6) (x+6)

(x - 5)^2

(2x + 7)^2

(3x - 1) (2x+ 5)

2(x -3) (x + 2)

-3(4x - 5) (2x - 1)

2(3x - 4)^2

## Factoring

ab + ac= a(b + c)

Decomposition example:

8x^2 + 22x + 15

To get 120 as the product multiply 8 by 15

P: 120

S: 22

Then find to numbers that multiply to 120 and add to 22 which in this case are 10 and 12

= 8x^2 + 10x + 12x + 15 Then factor it out

= 2x(4x + 5) + 3(4x + 5)

= (4x + 5) (2x + 3)

Factoring By grouping example:

ax - bx - ay + by

= x(a - b) - y(a - b)

= (a - b) (x - y)

xy - 4y + 3x - 12

= y(x - 4) + 3(x - 4)

= (x - 4) (y + 3)

Factor the following and use the correct way out of the three above:

- 3x - 15y
- 5pqr - pqs - 10pqt
- 2b^2 + 9b + 7
- 8am - 3bn - 6an + 4bm
- 6x^2 - 17x + 5
- 2 - 6b - 4c + 12bc
- 2x^2 - 10x - 12

## Difference of squares

x^2 - 9

= (x +3) (x-3)

a^4 - 1

= (a^2 + 1) (a^2 - 1)

= (a^2 + 1) (a + 1) ( a - 1)

Factor the following:

- x^2 - 25
- 49^2 - 81^2
- 5x^2 - 5
- a^2 - 81
- 25d^2 - 16

## Solve the following by completing the square

## Solveuse this equation for reference h= a(x - h)^2 + k - If the pistol bullet is fired vertically at an initial speed of 100 m/s, the height in meters after t seconds is given by
**h= 100t - 10t^2**
a) Find the maximum height attained by the bullet b) when does the bullet return to the ground? | ## Solve 2. A producer of synfuel from coal estimates that the cost in dollars per barrel for a production run of x thousand barrels is given by a) How many thousand barrels should be produced each run to minimize the cost per barrel? b) What is the minimum cost per barrel? | ## Solve 3. in the Olympics, a hammer was thrown according to the curve given by h= 0.975 - 4.9t^2 + 14.7t where h is the height reached by the hammer , in meters , and t was the time taken in seconds.a) How long did the hammer take to reach its maximum height? b) How high was it above the ground when it was thrown? |

## Solve

use this equation for reference h= a(x - h)^2 + k

- If the pistol bullet is fired vertically at an initial speed of 100 m/s, the height in meters after t seconds is given by
**h= 100t - 10t^2**

a) Find the maximum height attained by the bullet

b) when does the bullet return to the ground?

## Solve

2. A producer of synfuel from coal estimates that the cost in dollars per barrel for a production run of x thousand barrels is given by **C= 9x^2 - 180x + 940**

a) How many thousand barrels should be produced each run to minimize the cost per barrel?

b) What is the minimum cost per barrel?

## Solve

**h= 0.975 - 4.9t^2 + 14.7t**where h is the height reached by the hammer , in meters , and t was the time taken in seconds.

a) How long did the hammer take to reach its maximum height?

b) How high was it above the ground when it was thrown?

## Solve The equations using the quadratic formula

## Hints

Remember: 50x^2 + 20x + 2 = 0

a = 50

b = 20

c = 2

solve the following:

- x^2 + x - 6 = 0
- 6x^2 + 19x + 15 = 0
- 16x^2 - 9 = 0
- x^2 + 6x - 16 = 0
- 4x^2 - 20x + 25 = 0