Download Theory: The quadratic formula.

Learning Activity for MATH 105
College Algebra Spring 2013
NAME:
Section:
Theory: The quadratic formula.
We have just seen that the roots of an equation of the form:
ax2 + bx + c = 0
are found using the quadratic formula (found by completing the square).
√
−b ± b2 − 4ac
x=
.
2a
The Discriminant
For an equation of the form ax2 + bx + c = 0 the term
b2 − 4ac is called the discriminant.
• If the discriminant is zero then the equation has only One Real solution.
• If the discriminant is positive then the equation has Two Real solutions.
• If the discriminant is negative then the equation has No Real solution.
Solve 3x2 − 21 = 0
21
3
√
=⇒ x = ± 7
=⇒ x2 =
Solve: x2 − 3x − 4 = 0:
=⇒ (x − 4)(x + 1) = 0
=⇒ x − 4 = 0 or x + 1 = 0
=⇒ x = 4 or x = −1
Solve: 3x2 + 2x = 7.
=⇒ 3x2 + 2x − 7 = 0
a = 3, b = 2, and c = −7
p
p
√
√
−2 ± 4 − 4(3)(−7)
−2 ± 22 − 4(3)(−7)
−2 + 88
−2 − 88
=
=
and
x=
2(3)
6
6
6
Now consider solving the following equation x4 − 5x2 + 4 = 0.
Here a substitution needs to be maid (Let y = x2 :
=⇒ y 2 − 5y + 4 = 0
=⇒ (y − 4)(y − 1) = 0
=⇒ y = 4 or y = 1
=⇒ x2 = 4 or x2 = 1
=⇒ x = 2, or x = −2, or x = 1, or x = −1.
The Vertex of a Parabola
The x-coordinate of the vertex of a parabola is centered in between the roots. Thus, for the graph
of
f (x) = ax2 + bx + c
the vertex is located at:
−b
,f
2a
−b
2a
.
Note we find the x-value first, and substitute into the given function for the y − value.
Rocket Science:
A model rocket is launched with an initial velocity of 120 ft/sec from a height of 80 ft. The height
of the rocket t seconds after it has been launched is given by the function s(t) = −16t2 + 120t + 80.
Determine the time at which the rocket reaches its maximum height and find the maximum height.
The rocket reaches its maximum height when it is at its vertex. This happens when
t=
−b
= −120/(−32) = 3.75
2a
The height of the rocket is a maximum 3.75 seconds after its launch.
To find the height we only need to plug into the model:
s(3.75) = −16(3.752 ) + 120(3.75) + 80 = 305
The maximum height of the rocket is 305 feet.
Two Numbers
One number is 5 greater than another. The product of the numbers is 36. Find the numbers. Let
x be one of the numbers. Then x + 5 is the second number.
x(x + 5) = 36
=⇒ x2 + 5x − 36 = 0
=⇒ (x + 9)(x − 4) = 0
The two numbers are −9 and −4 or the two numbers are 4 and 9.