2. Derive the Quadratic Formula
 ax 2 + bx + c = 0

3. The Quadratic Formula
 The roots of the polynomial ax 2 + bx + c and the
solutions of the quadratic equation ax 2 + bx + c = 0
-b ± b2 - 4ac
are x = where a ¹ 0 and b2 - 4ac ³ 0
2a

4. Example 1 Find the roots of a polynomial
Find the roots of x2 – 6x + 3.
SOLUTION
The roots of x2 – 6x + 3 are the values of x for which
x2 – 6x + 3 = 0.
–b +
– b2 – 4ac
x= Quadratic formula
2a
– ( – 6) +
– ( – 6)2 – 4( 1 )( 3 ) Substitute values in the quadratic
x= formula: a = 1, b = – 6, and c = 3.
2( 1 )

5. Example 1 Find the roots of a polynomial
6 + 24
= – Simplify.
2
6 +2 6
= – Simplify radical.
2
2(3 + 6 )
–
= = 3 +
– 6 Divide out factor of 2.
2
ANSWER
The roots of x2 – 6x + 3 are 3 + 6 and 3 – 6.

6. Example 1 Find the roots of a polynomial
CHECK Substitute each root for x. The polynomial
should simplify to 0.
(3 + 6 )2 – 6 (3 + 6) + 3
= 9 + 6 6 + 6 – 18 – 6 6 + 3
= 0
(3 – 6 )2 – 6 (3 – 6) + 3
= 9 – 6 6 + 6 – 18 + 6 6 + 3
= 0

7. Example 2 Multiple Choice Practice
Which is one of the solutions to the equation
2x2 – 7 = x?
1 1
– 57 + 57
4 4
–1 + 57 1+ 57
4 4
SOLUTION
Write original
2x2 – 7 = x equation.
Write in standard
2x2 – x – 7 = 0 form.

8. Example 2 Multiple Choice Practice
–b+
– b2 – 4ac Quadratic formula
x =
2a
Substitute values
– ( – 1) + ( – 1)2 – 4( 2 )(–7) in the quadratic
– formula:
=
2( 2 ) a = 2, b = –1, and
c = –7.
1 + 57
–
= Simplify.
4
ANSWER
1 + 57
One solution is .
4
The correct answer is D.

9. Methods for Solving Quadratic Equations

10. Example 4 Choose a solution method
Tell what method(s) you would use to solve the quadratic
equation. Explain your choice(s).
a. 10x2 – 7 = 0
b. x2 + 4x = 0
c. 5x2 + 9x – 4 = 0
SOLUTION
a. The quadratic equation can be solved using square
roots because the equation can be written in the form x2
d. =

11. Example 4 Choose a solution method
b. The quadratic equation can be solved by factoring
because the expression x2 + can be factored easily.
4x
Also, the equation can be solved by completing the
square because the equation is of the form
where a ax2and b is an=even number.
1 + bx + c 0 =
c. The quadratic equation cannot be factored easily, and
completing the square will result in many fractions. So,
the equation should be solved using the quadratic
formula.

12. Example 3 Use the quadratic formula
FILM PRODUCTION
For the period 1971 – 2001, the number y of films
produced in the world can be modeled by the function
y = 10x2 – 94x + 3900 where x is the number of years
since 1971. In what year were 4200 films produced?
SOLUTION
y = 10x2 – 94x + 3900 Write function.
4200 = 10x2 – 94x + 3900 Substitute 4200 for y.
0 = 10x2 – 94x – 300 Write in standard form.

13. Example 3 Use the quadratic formula
Substitute values in the
– ( – 94) +
– ( – 94)2 – 4 (10)(–300) quadratic formula: a = 10,
x =
2(10) b = –94, and c = –300.
94 + 20,836
– Simplify.
=
20
94 + 20,836
The solutions of the equation are ≈ 12 and
20
94 – 20,836
≈ –3.
20
ANSWER
There were 4200 films produced about 12 years after 1971,
or in 1983.

14. 10.7 Warm-up (Day 1)
Use the quadratic formula to Find the roots
1. x 2 + 4x -1
Use the quadratic formula to solve the equation
2. x 2 - 8x +16 = 0
3. x 2 - 5x = -21
4. 4z 2 = 7z + 2

15. 10.7 Warm-up (Day 2)
Use the quadratic formula to solve the equation
1. n2 +1 = 5n
2. 2z + 4 = 3z 2

16. 10.7 Warm-up (Day 3)
Use the quadratic formula to solve the equation
1. 4x 2 + 6x = 3x 2 - 4x +1
2. 7r 2 - 2r = 6