section 7.3 using the quadratic formula to find real solutions
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Section 7.3
Using the Quadratic Formula to Find Real Solutions
7.3 Lecture Guide: Using the Quadratic Formula to Find Real Solutions
Objective 1: Use the quadratic formula to solve quadratic equations with real solutions.
Methods for solving quadratic equations we have covered so far:•Tables --- the zeros in a table correspond to the solutions
•Graphs --- the x-intercepts correspond to the solutions
•Factoring --- the factors of the polynomial correspond to the solutions
•Extraction of roots
•Completing the square
The quadratic formula can be used to solve any quadratic equation --- an all purpose tool. The quadratic formula can be derived using completing the square as shown below.
Solving by Completing the Square 2 0ax bx c
Step 1. Write the equation with the constant term on the right side.
Step 2. Divide both sides of the equation by the coefficient of to obtain a coefficient of 1 for .
2 0ax bx c 2ax bx
2x
2x2ax bxa a
2 bx x
a
Step 3. Take one-half of the coefficient of x, square this number, and add the result to both sides of the equation.
Step 4. Write the left side of the equation as a perfect square.
Solving by Completing the Square 2 0ax bx c
2 bx x
a
2
2b
xa
Solving by Completing the Square 2 0ax bx c
Step 5. Solve this equation by extraction of roots.
2b
xa
2 42 2b b ac
xa a
2 42
b b acx
a
The Quadratic Formula, which gives the solutions of the quadratic equation with real coefficientsa, b, and c, when is:
2 0ax bx c 0a
Use the quadratic formula to determine the exact solutions of each quadratic equation. Then approximate each solution to thenearest hundredth.
1. 2 2 2 0x x
Use the quadratic formula to determine the exact solutions of Each quadratic equation. Then approximate each solution to thenearest hundredth.
2. 24 6 1 0x x
Use the quadratic formula to determine the exact solutions of Each quadratic equation. Then approximate each solution to thenearest hundredth.
3. 22 3 3x x
Use the quadratic formula to determine the exact solutions of Each quadratic equation. Then approximate each solution to thenearest hundredth.
4. 2 3 1x x
A part of the quadratic formula that determines the nature of the solutions is the expression under the radical symbol. is called the discriminant. This expression is important since
is not a real number if is ______________.
Objective 2: Use the discriminant to determine the nature of the solutions of a quadratic equation.
2 4b ac2 4b ac
2 4b ac
There are three possibilities for the solutions of .
Value of the Discriminant
Solutions of The Parabola
1. Two distinct real solutions
Two x-intercepts
The Nature of the Solutions of a Quadratic Equation
2 0ax bx c
2 0ax bx c 2y ax bx c
2 4 0b ac
-4
8
-4 4
x
y
Graphical Example
2 2y x x
There are three possibilities for the solutions of .
Value of the Discriminant
Solutions of The Parabola
2. A double realsolution
One x-intercept with the vertex on the x-axis
2 4 0b ac
2 0ax bx c
Graphical Example
-4 4
-4
8
x
y
2 4 4y x x
2 0ax bx c 2y ax bx c
There are three possibilities for the solutions of .
Value of the Discriminant
Solutions of The Parabola
3. Neither solution is real; both solutions are complex numbers with imaginary parts. These solutions will be complex conjugates.*
No x-intercepts
* Complex numbers are covered in Section 7.5.
2 4 0b ac
2 0ax bx c
-4
8
-4 4
x
y
2 1y x
Graphical Example
2y ax bx c 2 0ax bx c
Compute the value of each discriminant, , and determine the nature of the solutions.
2 4b ac
5. Equation
Discriminant Nature of Solutions Graph
5,5,1 by 5,5,1
23 5 2 0x x
Compute the value of each discriminant, , and determine the nature of the solutions.
2 4b ac
6. Equation
Discriminant Nature of Solutions Graph
5,5,1 by 5,5,1
24 12 9 0x x
Compute the value of each discriminant, , and determine the nature of the solutions.
2 4b ac
7. Equation
Discriminant Nature of Solutions Graph
2 6 10 0x x
5,5,1 by 5,5,1
8. Use the quadratic formula to determine the exact solutions of the quadratic equation . Then approximate each solution to the nearest hundredth.
2 4 0x x
(a)
(b)
5,5,1 by 5,5,1
9. Use the graph and the solution from problem 8 to solve the following inequalities.
2 4 0x x
2 4 0x x
10. Use the quadratic formula to determine the exact solutions of the quadratic equation . Then approximate each solution to the nearest hundredth.
2 4 1 0x x
(a)
(b)
11. Use the graph and the solution from problem 10 to solve the following inequalities.
2 4 1 0x x
2 4 1 0x x 8,10,1 by 8,8,1
(a) Determine the overhead costs for the company. Hint: Evaluate .
22 50 20P x x x
0P
0P x
12. The weekly profit in dollars for selling x bottles of hand lotion is given by .
(b) Determine the break-even values for the company. Hint: Determine to the nearest unit when .
4 52
x 4 5
2x
13. Construct a quadratic equation in x that has solutions of
and .