Quadratic Equations

Review ^2 means to the power of 2

Expanding


To expand binomials collect like terms to simplify
(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



  1. (x - 4) (x + 7)

  2. (x - 6) (x+6)

  3. (x - 5)^2

  4. (2x + 7)^2

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

  6. 2(x -3) (x + 2)

  7. -3(4x - 5) (2x - 1)

  8. 2(3x - 4)^2

Factoring

If every term or a polynomial is divisible by the same constant, the constant is called a common factor

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:


  1. 3x - 15y
  2. 5pqr - pqs - 10pqt
  3. 2b^2 + 9b + 7
  4. 8am - 3bn - 6an + 4bm
  5. 6x^2 - 17x + 5
  6. 2 - 6b - 4c + 12bc
  7. 2x^2 - 10x - 12

Difference of squares

Example:

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:

  1. x^2 - 25
  2. 49^2 - 81^2
  3. 5x^2 - 5
  4. a^2 - 81
  5. 25d^2 - 16

Solve the following by completing the square

Solve The equations using the quadratic formula

Hints

Quadratic Equation: (-b±√(b^2-4ac))/2a

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

a = 50

b = 20

c = 2


solve the following:

  1. x^2 + x - 6 = 0
  2. 6x^2 + 19x + 15 = 0
  3. 16x^2 - 9 = 0
  4. x^2 + 6x - 16 = 0
  5. 4x^2 - 20x + 25 = 0