Solving Quadratic Equations: Complete Square, Formula, Applications

1. Chapter 5 Quadratics: Make connections between different representations of the quadratic function. You will also solve the quadratic equation using multiple methods.

2. 5.2 Solving Quadratics Students will solve quadratics by completing the square and quadratic formula. Students will determine the number of solutions without solving a quadratic equation. Students will solve real world applications.

3. 5.2.1 Perfect Square Quadratics This will be the vertex form for a quadratic equation, having the binomial squared We will discuss how to solve these instead of factoring Work through 5-60 on page 282 You would have noticed with factored form answers are always integers With perfect squares we can write our answers in simplest radical form

4. Solving vertex form • Make sure equation is set equal to zero • Move the k term to the other side of the equals sign • Divide by a • Square root both sides, this gives you a plus and minus • Move the h term to the other side

5. Number of solutions When solving vertex form how do you know the number of solutions you will have 2 solutions If when you take the square root the number underneath is positive 1 solution If when you take the square root the number underneath is zero, or k is zero 0 solutions If when you take the square root the number underneath is negative

6. Examples

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8. 5.2.2 Completing the Square This method takes the standard form of a quadratic and rewrites it in vertex form We will discuss the process when a=1 and when a is not 1

9. When a=1 • Write quadratic in order • Group first 2 terms • Take b/2 and square it • Add answer from step 3 inside the ( ) • Subtract answer from step 3 outside the ( ) • The trinomial inside the ( ) is a perfect square so factor it • Combine terms outside ( )

10. Example

11. When a is not 1 Given the following problem what do you think you would do, can you just manipulate the previous steps

12. Process Steps are very similar 2 extras • Write quadratic in order • Group first 2 terms • Factor out a • Take new b/2 and square it • Add answer from step 4 inside the ( ) • Multiple answer from step 4 by a and subtract outside ( ) • The trinomial inside the ( ) is a perfect square so factor it • Combine terms outside ( )

13. Example again

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15. 5.2.4 Quadratic Formula Try solving the following problem by • Zero Product Rule • Completing the square You can skip a step by using the quadratic formula which is the process of taking standard form rewriting it in factored form and then solving it Watch

16. Quadratic Formula

17. Examples Solve the following by using the quadratic formula

18. What did you notice How can we tell the number of solutions when using the quadratic formula, this is called the discriminant 2 solutions if value under radical is positive 1 solution if value under radical is zero 0 solutions if value under the radical is negative

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20. 5.2.5 Solving Quadratics Work in groups Page 299 5-108 5-109 5-110

21. Word Problems Key terms and ideas to remember Max height of something would be the vertex Initial height of something would be the y-intercept Max distance is the x-intercept When given a distance and you need to calculate the height sub value in for x and solve When given the height and you need to calculate the distance sub in value for y and solve for x

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23. 5.2.6 Introducing Complex Numbers This is the start of imaginary numbers Square root of -1 is i A complex number is a combination of a real number and an imaginary number a+bi We see these complex numbers when the graph of a quadratic doesn’t cross the x axis

24. General Rules

25. Examples Pg 306 Work through 5-127 and 5-128

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