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Mastering Quadratic Equations: The Versatile Quadratic Formula

Learn how to solve quadratic equations using the quadratic formula, understand the discriminant, and apply equations to real-world scenarios. Step-by-step examples and graphing techniques included.

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Mastering Quadratic Equations: The Versatile Quadratic Formula

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  1. Chapter 8 Quadratic Functions

  2. Chapter Sections 8.1 – Solving Quadratic Equations by Completing the Square 8.2 – Solving Quadratic Equations by the Quadratic Formulas 8.3 – Quadratic Equations: Applications and Problem Solving 8.4 – Writing Equations in Quadratic Form 8.5 – Graphing Quadratic Functions 8.6 – Quadratic and Other Inequalities in One Variable

  3. Solving Quadratic Equations by the Quadratic Formula § 8.2

  4. Quadratic Formula The quadratic formulacan be used to solve any quadratic equation. It is the most versatile method of solving quadratic equations.. To use the quadratic formula, the equation must be written in standard form ax2 + bx + c = 0, where a is the coefficient of the squared term, b is the coefficient of the first-degree term, and c is the constant. 3x2 + 4x – 5 = 0 a = 3, b =4, and c = – 5

  5. Quadratic Formula To Solve a Quadratic Equation by the Quadratic Formula Write the quadratic equation in standard form, ax2 + bx + c = 0, and determine the numerical values for a, b, and c. Substitute the values for a, b, and c into the quadratic formula and then evaluate the formula to obtain the solution.

  6. Quadratic Formula Example Solve x2 + 2x – 8 = 0 by using the quadratic formula. continued

  7. Quadratic Formula or

  8. Determine a Quadratic Equation Given Its Solutions Example Determine an equation that has the solutions -5 and 1.

  9. Discriminant Discriminant The discriminant of a quadratic equation is the expression under the radical sign in the quadratic formula.

  10. Solutions of a Quadratic Equation Solutions of a Quadratic Equation • For a quadratic equation of the form ax2 + bx + c = 0, a ≠ 0: • If b2 – 4ac > 0, the quadratic equation has two distinct real number solutions. • If b2 – 4ac = 0, the quadratic equation has a single real number solution. • If b2 – 4ac < 0, the quadratic equation has no real number solution.

  11. y y x x Discriminant Graphs of f(x) = ax2 + bx + c • If b2 – 4ac > 0, f(x) has twodistinct x-intercepts. or

  12. y y x x Discriminant Graphs of f(x) = ax2 + bx + c 2. If b2 – 4ac =0, f(x) has one single x-intercept. or

  13. y y x x Discriminant Graphs of f(x) = ax2 + bx + c 3. If b2 – 4ac <0, f(x) has no x-intercept. or

  14. Study Applications That Use Quadratic Equations Example Mary Olson owns a business that sells cell phones. The revenue, R(n), from selling the cells phones is determined by multiplying the number of cell phones by the selling price per phone. Suppose the revenue from selling n cell phone, n ≤ 50, is R(n)=n(50-0.2n) where (50-0.2n) is the price per cell phone, in dollars. Find the revenue when 30 cell phones are sold. continued

  15. Study Applications That Use Quadratic Equations To find the revenue when 30 cell phones are sold, we evaluate the revenue function for n = 30. The revenue from selling 30 cell phones is $1,320.

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