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
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
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