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Solving Quadratic Equations Algebraically

2.2. Solving Quadratic Equations Algebraically. Objectives: Solve equations by: Factoring Square Root of Both Sides Completing the Square Quadratic Formula Solve equations in quadratic form. Definition of a Quadratic Equation. An equation that can be written in the form

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Solving Quadratic Equations Algebraically

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  1. 2.2 Solving Quadratic Equations Algebraically Objectives: Solve equations by: Factoring Square Root of Both Sides Completing the Square Quadratic Formula Solve equations in quadratic form.

  2. Definition of a Quadratic Equation An equation that can be written in the form ax2 + bx + c = 0 with real constants a, b, & c, with a ≠ 0. Techniques to solve a quadratic equation: • Techniques that sometimes work Factoring Taking the square root of both sides of an equation • Techniques that always work Completing the Square Using the Quadratic Formula

  3. Example #1aSolving by Factoring Solve by Factoring:

  4. Example #1bSolving by Factoring Solve by Bottoms Up Method: Don’t forget to first factor out the GCF, if necessary. To factor this trinomial you must first multiply a by c, factor normally, divide both factors by a & reduce, and finally bring “bottoms up”.

  5. Example #2Solving ax2 = b Solve by Taking the Square Root of Both Sides:

  6. Example #3Solving a(x−h)2 = k Solve by Taking the Square Root of Both Sides:

  7. Example #4Solving a Quadratic Equation by Completing the Square Solve by Completing the Square: Completing the square only works when the coefficient of x2 is a 1. Always divide every term by the leading coefficient before attempting to complete the square.

  8. Example #4Solving a Quadratic Equation by Completing the Square Solve by Completing the Square: Remember after completing the square, add it to both sides of the equation.

  9. The Quadratic Formula

  10. Example #5Solving a Quadratic Equation by Using the Quadratic Formula Solve by Using the Quadratic Formula: Quadratic equations must be in Standard Form ax2 + bx + c = 0 before using the Quadratic Formula.

  11. The Discriminant The discriminant is used to determine the number of solutions without solving the problem.

  12. Example #6Determining the Number of Solutions by Using the Discriminant Find the Number of Solutions: −31 is NOT a solution to the equation. Since the discriminant is negative, this means that there is no real solution to the equation.

  13. Example #7Polynomials in Quadratic Form Solving an Equation in Quadratic Form: Compare this polynomial equation with the quadratic form: ax2+ bx + c = 0 This polynomial can be factored using Bottoms Up!

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