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Solving Equations: Graphical, Numerical, and Algebraic Techniques for Quadratics

This chapter covers essential techniques for solving equations, focusing on graphical, numerical, and algebraic methods for quadratic equations. You will learn how to solve equations graphically using graphing utilities, approximate solutions numerically with tables, and find intersections. Key concepts include quadratic equations, completing the square, the quadratic formula, and identifying points of intersection. Through examples, you will gain practical experience in solving these equations using various methods, enhancing your understanding of solving techniques in mathematics.

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Solving Equations: Graphical, Numerical, and Algebraic Techniques for Quadratics

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  1. Chapter P:Prerequisite Information Section P-5: Solving Equations Graphically, Numerically, and Algebraically

  2. Objectives • You will learn about: • Solving equations graphically • Solving quadratic equations • Approximating solutions of equations graphically • Approximating solutions of equations numerically with tables • Solving equations by finding intersections • Why: • These basic techniques are involved in using a graphing utility to solve equations

  3. Vocabulary • Quadratic equation in x • Completing the square • Quadratic formula • Point of intersection • Discriminant

  4. Example 1:Solving by Finding x-intercepts • Solve the equation 2x2 – 3x – 2 = 0. • Solve graphically using the trace key on your calculator. • Check your work algebraically.

  5. Zero Factor Property • Let a and b be real numbers. • If ab = 0, then either a = 0 or b = 0

  6. Quadratic Equation in x • A quadratic equation in x is one that can be written in the form: • ax2 + bx + c = 0 • where a, b, and c are real numbers and a ≠ 0

  7. Example 2:Solve by extracting square roots • Solve (2x – 1)2 = 9 algebraically.

  8. Completing the Square • To solve ax2 + bx + c = 0 by completing the square, add (b/2)2 to both sides of the equation and factor the left side of the new equation.

  9. Example 3:Solve by Completing the Square • Solve 4x2 - 20x + 17 = 0 by completing the square.

  10. Quadratic Formula • The solutions of the quadratic equation ax2 + bx + c = 0 where a ≠ 0 are given by the quadratic formula:

  11. Example 4:Solve using the quadratic formula • Solve the equation 3x2 – 6x = 5

  12. Solving Quadratic Equations Algebraically • There are four basic ways to solve quadratic equations algebraically: • Factoring • Extracting square roots • Completing the square • Using the quadratic formula

  13. Examples 5 and 6:Solve Graphically and Using Tables • Solve the equation x3 – x – 1 = 0

  14. Agreement About Approximate Solutions: • For approximations, round to a value that is reasonable for the context of the problem. For all others, round to two decimal places unless directed otherwise.

  15. Example 7:Solve by Finding Intersections • Solve the equation |2x – 1|= 6 • First graphically • Check algebraically

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