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 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.

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

Step 5. Solve this equation by extraction of roots.

The Quadratic Formula, which gives the solutions of the quadratic equation with real coefficients a, b, and c, when is:

Use the quadratic formula to determine the exact solutions of each quadratic equation. Then approximate each solution to the nearest hundredth. 1.

Use the quadratic formula to determine the exact solutions of Each quadratic equation. Then approximate each solution to the nearest hundredth. 2.

Use the quadratic formula to determine the exact solutions of Each quadratic equation. Then approximate each solution to the nearest hundredth. 3.

Use the quadratic formula to determine the exact solutions of Each quadratic equation. Then approximate each solution to the nearest hundredth. 4.

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.

There are three possibilities for the solutions of. Value of the Discriminant Solutions ofThe Parabola 1.Two distinct real solutions Two x-intercepts The Nature of the Solutions of a Quadratic Equation Graphical Example

There are three possibilities for the solutions of. Value of the Discriminant Solutions ofThe Parabola 2.A double real solution One x-intercept with the vertex on the x- axis Graphical Example

There are three possibilities for the solutions of. Value of the Discriminant Solutions ofThe 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. Graphical Example

Compute the value of each discriminant,, and determine the nature of the solutions. 5. Equation DiscriminantNature of SolutionsGraph

Compute the value of each discriminant,, and determine the nature of the solutions. 6. Equation Discriminant Nature of SolutionsGraph

Compute the value of each discriminant,, and determine the nature of the solutions. 7. Equation DiscriminantNature of SolutionsGraph

8. Use the quadratic formula to determine the exact solutions of the quadratic equation. Then approximate each solution to the nearest hundredth.

(a) (b) 9. Use the graph and the solution from problem 8 to solve the following inequalities.

10. Use the quadratic formula to determine the exact solutions of the quadratic equation. Then approximate each solution to the nearest hundredth.

(a) (b) 11. Use the graph and the solution from problem 10 to solve the following inequalities.

(a) Determine the overhead costs for the company. Hint: Evaluate. 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.

13. Construct a quadratic equation in x that has solutions of and.