Intro: We already know the standard form of a quadratic

1
The Quadratic Formula

• Intro We already know the standard form of a
  quadratic equation is
• y ax2 bx c
• The constants are a , b, c
• The variables are y, x

2
What it means

• The ROOTS (or solutions) of a polynomial are its
  x-intercepts
• Recall The x-intercepts occur where y 0.

3
The Easy Way

• Example Find the roots y x2 x - 6
• Solution Factoring y (x 3)(x - 2)
• 0 (x 3)(x - 2)
• The roots are
• x -3 x 2

4
The Challenge

• But what about NASTY trinomials that dont
  factor?
• Abu Ja'far Muhammad ibn Musa Al-Khwarizmi
• Born about 780 in Baghdad (Iraq)
• Died about 850

5
The Formula

• After centuries of work, mathematicians realized
  that as long as you know the coefficients, you
  can find the roots of the quadratic. Even if it
  doesnt factor!

6
Example
7
Solution!
8
Check your answer!
Plug in your answers for x.
If youre right, youll get y 0.
9
Your turn
10
Tricks of the Trade
Remember All the terms must be on one side
BEFORE you use the quadratic formula.

• Example Solve 3m2 - 8 10m
• Solution 3m2 - 10m - 8 0
• a 3, b -10, c -8

11
Your turn!

• Solve 3x2 7 - 2x
• Solution 3x2 2x - 7 0
• a 3, b 2, c -7

12
Weird History!
We use the quadratic formula to solve second
degree equations. Mathematicians tried for 300
years to solve higher-degree equations until
Niels Abel (top picture) proved that no formula
can be used to solve all fifth-degree equations.
He was 22!

• Evariste Galois (bottom picture) showed that
  there is no universal formula for any equations
  higher than the fourth degree. When Galois was
  20, he wrote in ONE NIGHT much of the basis for a
  new theory of solving equations. Sadly, he was
  killed in a duel the next day.
• MORAL Dont do your homework late at night.