Practice and Problem-Solving Exercises
A Practice
See Problem 1.
Solve each equation using the Quadratic Formula.
-
x
2
−
4
x
+
3
=
0
x squared , minus 4 x plus 3 equals 0
-
x
2
+
8
x
+
12
=
0
x squared , plus 8 x plus 12 equals 0
-
2
x
2
+
5
x
=
7
2 x squared , plus 5 x equals 7
-
3
x
2
+
2
x
−
1
=
0
3 x squared , plus 2 x minus 1 equals 0
-
x
2
+
10
x
=
−
25
x squared , plus 10 x equals negative 25
-
2
x
2
−
5
=
−
3
x
2 x squared , minus 5 equals negative 3 x
-
x
2
=
3
x
−
1
x squared , equals 3 x minus 1
-
6
x
−
5
=
−
x
2
6 x minus 5 equals negative , x squared
-
3
x
2
=
2
(
2
x
+
1
)
3 x squared , equals 2 open 2 x plus 1 close
-
2
x
(
x
−
1
)
=
3
2 x open x minus 1 close equals 3
-
x
(
x
−
5
)
=
−
4
x open x minus 5 close equals negative 4
-
12
x
+
9
x
2
=
5
12 x plus , 9 x squared , equals 5
See Problem 2.
-
Fundraising Your class is selling boxes of flower seeds as a fundraiser. The total profit p depends on the amount x that your class charges for each box of seeds. The equation
p
=
−
0
.
5
x
2
+
25
x
−
150
p equals negative 0 . 5 , x squared , plus 25 x minus 150 models the profit of the fundraiser. What's the smallest amount, in dollars, that you can charge and make a profit of at least $125?
-
Baking Your local bakery sells more bagels when it reduces prices, but then its profit changes. The function
y
=
−
1000
x
2
+
1100
x
−
2.5
y equals negative , 1000 , x squared , plus , 1100 , x minus 2.5 models the bakery's daily profit in dollars, from selling bagels, where x is the price of a bagel in dollars. What's the highest price the bakery can charge, in dollars, and make a profit of at least $200?
See Problem 3.
Evaluate the discriminant for each equation. Determine the number of real solutions.
-
x
2
+
4
x
+
5
=
0
x squared , plus 4 x plus 5 equals 0
-
x
2
−
4
x
−
5
=
0
x squared , minus 4 x minus 5 equals 0
-
−
4
x
2
+
20
x
−
25
=
0
negative 4 , x squared , plus 20 x minus 25 equals 0
-
−
2
x
2
+
x
−
28
=
0
negative 2 , x squared , plus x minus 28 equals 0
-
2
x
2
+
7
x
−
15
=
0
2 x squared , plus 7 x minus 15 equals 0
-
6
x
2
−
2
x
+
5
=
0
6 x squared , minus 2 x plus 5 equals 0
-
−
2
x
2
+
7
x
=
6
negative 2 , x squared , plus 7 x equals 6
-
x
2
−
12
x
+
36
=
0
x squared , minus 12 x plus 36 equals 0
-
x
2
+
8
x
=
−
16
x squared , plus 8 x equals negative 16
-
3
x
2
+
x
=
−
3
3 x squared , plus x equals negative 3
-
x
+
2
=
−
3
x
2
x plus 2 equals negative 3 , x squared
-
12
x
(
x
+
1
)
=
−
3
12 x open x plus 1 close equals negative 3
See Problem 4.
-
Business The weekly revenue for a company is
r
=
−
3
p
2
+
60
p
+
1060
,
r equals negative 3 , p squared , plus 60 p plus , 1060 , comma where p is the price of the company's product. Use the discriminant to find whether there is a price for which the weekly revenue would be $1500.
-
Physics The equation
h
=
80
t
−
16
t
2
h equals 80 , t minus . 16 t squared models the height h in feet reached in t seconds by an object propelled straight up from the ground at a speed of 80 ft/s. Use the discriminant to find whether the object will ever reach a height of 90 ft.
B Apply
-
Think About a Plan The area of a rectangle is
36
in
.
2
.
36 , in , . squared , . The perimeter of the rectangle is 36 in. What are the dimensions of the rectangle to the nearest hundredth of an inch?
- How can you write an equation using one variable to find the dimensions of the rectangle?
- How can the discriminant of the equation help you solve the problem?
-
Writing Summarize how to use the discriminant to analyze the types of solutions of a quadratic equation.
