2. Quadratic Equations, Inequalities, and Functions
Chapter 10
Quadratic Equations, Inequalities,
and Functions

3. 10.2
The Quadratic Formula

4. Derive the quadratic formula.
Objectives
Derive the quadratic formula.
Solve quadratic equations using the quadratic formula.
Use the discriminant to determine the number and type of solutions.
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5. Derive the Quadratic Formula
We can use the method of completing the square to solve any quadratic equation. However, completing the square can become tedious and time consuming. In this section, we begin with the general quadratic equation
By applying the method of completing the square to this general equation, we can develop a formula for finding the solution of any specific quadratic equation.
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6. Derive the Quadratic Formula
Applying the Method of Completing the Square
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7. Derive the Quadratic Formula
Applying the Method of Completing the Square Cont’d.
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8. Derive the Quadratic Formula
Applying the Method of Completing the Square Cont’d.
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9. Stating and Applying the Quadratic Formula
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10. Stating and Applying the Quadratic Formula
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11. Statement of the Quadratic Formula
You should always check each solution in the original equation.
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12. Statement of the Quadratic Formula
Always try factoring first. If factoring is difficult or impossible, use the quadratic equation.
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13. Using Quadratic Formula (Irrational Solutions)
10.2 The Quadratic Formula
Using Quadratic Formula (Irrational Solutions)
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14. Using Quadratic Formula (Irrational Solutions)
10.2 The Quadratic Formula
Using Quadratic Formula (Irrational Solutions)
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15. Simplifying the Results Using the Quadratic Formula
Be sure to factor first; then divide out the common factor.
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16. 10.2 The Quadratic Formula
Using Quadratic Formula (Nonreal Complex Solutions)
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17. 10.2 The Quadratic Formula
Using Quadratic Formula (Nonreal Complex Solutions)
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18. 10.2 The Quadratic Formula
Using the Discriminant to Predict Number and Type of Solutions
Discriminant
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19. What Information the Discriminant Predicts
10.2 The Quadratic Formula
What Information the Discriminant Predicts
If the discriminant is a perfect square (including 0), then the equation can be factored; otherwise, the quadratic formula should be used.
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20. What Information the Discriminant Predicts
10.2 The Quadratic Formula
What Information the Discriminant Predicts
The value of the discriminant is –44 and the equation has two nonreal complex solutions and is best solved with the quadratic formula.
Because the discriminant is 0, there is only one rational solution, and the equation can be solved by factoring.
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21. What Information the Discriminant Predicts
10.2 The Quadratic Formula
What Information the Discriminant Predicts
The discriminant is a perfect square so there will be two rational solutions and the equation can be solved by factoring.
The value of the discriminant is 109 and the equation has two irrational solutions and is best solved with the quadratic formula.
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