Derive the Quadratic
Formula
Focus 10 Learning Goal –
(HS.A-REI.B.4, HS.F-IF.B.4, HS.F-IF.C.7, HS.F-IF.C.8)
=
Students will sketch graphs of quadratics using key features and solve quadratics using the
quadratic formula.
4
3
2
1
In addition to
level 3, students
make
connections to
other content
areas and/or
contextual
situations
outside of math.
Students will
sketch graphs of
quadratics using
key features and
solve quadratics
using the
quadratic
formula.
- Students will be
able to write,
interpret and
graph quadratics
in vertex form.
Students will be
able to use the
quadratic
formula to solve
quadratics and
are able to
identify some key
features of a
graph of a
quadratic.
Students
will have
partial
success at
a 2 or 3,
with help.
0
Even with
help, the
student is
not
successful
at the
learning
goal.
Solve for x by completing the square:
x2 + 2x – 8 = 0
1. Add 8 to both sides:
1. x2 + 2x + ___ = 8
5. Square root both sides:
1. x + 1 = 3, -3
2. (1/2b)2 = (1/2 • 2)2 = 1
6. Get x alone:
3. Add 1 to both sides.
1. x = 2 and -4
1. x2 + 2x + 1 = 9
4. Factor:
1. (x + 1)2 = 9
7. Note: This equation could
have been solved by
factoring.
Solve for x by completing the square:
7x2 - 12x + 4 = 0
1. Divide all terms by the
leading coefficient.
1. 7/7x2 – 12/7x + 4/7 = 0
2. Subtract
4/
7
on both sides:
1. x2 - 12/7x + ___ = - 4/7
3. (1/2b)2 = (1/2 • 12/7)2 = 36/49
4. Add 36/49 to both sides.
1. x2 - 12/7x + 36/49 = - 4/7 + 36/49
2. x2 - 12/7x + 36/49 = 8/49
5. Factor:
1. (x - 6/7)2 = 8/49
6. Square root both sides:
1. x - 6/7 =
7. Get x alone:
1. x =
Reflection:
Which of these two problems makes more sense
to solve by completing the square?
Which makes more sense to solve by factoring?
How could you tell early in the problem solving
process which strategy to use?
How would you…
How would you solve this equation for x: ax + b = 0
(where a & b could be replaced by any number)?
Subtract b: ax = -b
Divide by a: x = -b/a
We could say that – b/a is a “formula” to solve ANY
equation in the form ax + b = 0.
Try it: 2x + 7 = 0
x = -7/2
How would you…
 How would you solve this equation for x: ax2 + bx + c = 0
(where a, b & c could be replaced by any number)?
 Since we don’t know if this will factor and leave rational
answers, we could use the method of completing the
square to get x alone.
 What would this look like since we don’t know the value
of a?
The Quadratic Formula!
𝒙 =
−𝒃± 𝒃𝟐 −𝟒𝒂𝒄
𝟐𝒂
You will need to memorize this!
Don’t worry, there is a catchy song to help you
with this !
Singing Time!
Practice using the quadratic formula instead of
factoring or completing the square to solve
x2 + 5x – 14 = 0.
 (5)  5  41 14
x
2 1
2
 5  25  56

2
59

2
2
 5  81

2
59

2
x = 2,-7
59

2
7
Practice using the quadratic formula instead
of factoring or completing the square to solve
x2 + 2x – 5 = 0.

2

22 6

2
2  4(1)( 5)
2(1)
2
2
4  20
2
2

2
24
AND
22 6

2
Solve these quadratic equations, using a different method for
each: solve by factoring, solve by completing the square and
solve by using the quadratic formula. Before starting, decide which
method you will use on each one.
1. 2x2 + 5x – 3 = 0
This one could be
factored to solve.
x = ½ or -3
2. x2 + 3x – 5 = 0
3. ½x2 - x – 4 = 0
This one could have
been solved by using
the quadratic
formula or by
completing the
square.
This one could
have been
factored to solve.
Because of the
leading coefficient,
you may have tried
a different strategy.
x=
x = 4 or -2
When do you factor, complete the
square, or use the quadratic formula?
 When is completing the square the most efficient method to use for
solving a quadratic equation?
 When it is not possible (or it is very difficult) to factor the quadratic expression
and when the leading coefficient and linear term are easy to deal with.
 When is the quadratic formula best?
 When it is not possible (or it is very difficult) to factor the quadratic expression,
and when the leading coefficient and/or linear term coefficient are fractions
that are not easily eliminated.
 When is factoring the most efficient method to use for solving a quadratic
equation?
 When the factors of the equation are obvious or fairly easy to find. When
factoring out the GCF or eliminating any fractional coefficients is possible.