Download Activity 2.2.6 Deriving the Quadratic Formula

Name:
Date:
Page 1 of 5
Activity 2.2.6 Deriving the Quadratic Formula
Directions: In Column 1, you are given precise instructions on the steps you need to take in
order to “complete the square” to solve the quadratic equation below. In Column 2, show the
manipulation you are making to the quadratic equation using numbers. In Column 3, show the
exact same manipulation you perform in Column 2, but to a quadratic equation in standard form
with the variables a, b, and c.
Solve the quadratic equation 5x 2 + 6x - 2 = 0 .
Column 1: Steps to Complete
the Square
Step 1: Isolate the x terms by
using the Addition Property
of Equality.
Column 2: Show with
Numbers for 5x 2 + 6x - 2 = 0
Column 3: Show with
Variables for ax 2 + bx + c = 0
Step 2: Change the
coefficient of the quadratic
term to 1 by dividing both
sides of the equation by the
coefficient of the quadratic
term.
Step 3: Divide the coefficient
of the linear term in half and
square it. Add the result to
both sides of the equation
using the Addition Property
of Equality. This creates a
perfect square trinomial,
which can be factored into the
square of a binomial.
Activity 2.2.6
Connecticut Core Algebra 2 Curriculum Version 3.0
Name:
Date:
Page 2 of 5
Step 4: Factor the perfect
square trinomial you just
created into the square of a
binomial.
Find a common denominator
to add together the rational
expressions.
Take the square root of both
sides of the equation and
simplify.
Solve for x and simplify the
expression.
Assume a > 0.
You have just derived the quadratic formula!!!
Activity 2.2.6
Connecticut Core Algebra 2 Curriculum Version 3.0
Name:
Date:
Page 3 of 5
NOTE: A quadratic equation in standard form is expressed as ax 2 + bx + c = 0 . For the given
quadratic equation a = 5, b = 6 and c = -2 . Substitute these values into the quadratic formula
and simplify your result below.
Does your result match the result in the last box of column two?
Applying the Quadratic Formula
x=
-b ± b2 - 4ac
2a
Directions: Now that you have derived the quadratic formula, use it to solve the following
quadratic equations below. First, identify the coefficients a, b, and c in the spaces given. Then
substitute those values into the quadratic formula and simplify the expression.
1.
x 2 +10x + 25 = 0
a = ______ b = ______ c = ______
Activity 2.2.6
2.
x 2 - 4x + 4 = 0
a = ______ b = ______ c = ______
Connecticut Core Algebra 2 Curriculum Version 3.0
Name:
3.
Date:
x2 - x - 6 = 0
a = ______ b = ______ c = ______
5.
6x 2 +11x + 3 = 0
a = ______ b = ______ c = ______
7.
2x 2 + 8x + 3 = 0
a = ______ b = ______ c = ______
Activity 2.2.6
Page 4 of 5
4.
x 2 - 2x - 35 = 0
a = ______ b = ______ c = ______
6.
5x 2 - 21x + 4 = 0
a = ______ b = ______ c = ______
8.
3x 2 - 6x -10 = 0
a = ______ b = ______ c = ______
Connecticut Core Algebra 2 Curriculum Version 3.0
Name:
Date:
Page 5 of 5
Recall that the square root of a negative number in not a Real Number. Therefore, if your
quadratic formula results with a negative number inside the square root symbol, then the answer
is “no Real Number solution”. Some of the following equations have a real number solution, and
some do not:
9. 2 x 2  x  3  0
10.
a = ______ b = ______ c = ______
11. x 2  9  0
a = ______ b = ______ c = ______
2x2  x  3  0
a = ______ b = ______ c = ______
12.
x2  9  0
a = ______ b = ______ c = ______
13. 5 x 2  8 x  5  0
a = ______ b = ______ c = ______
Activity 2.2.6
Connecticut Core Algebra 2 Curriculum Version 3.0