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Solving Quadratic Equations using Square Roots

Learn how to solve quadratic equations by finding square roots. Understand the properties of square roots and how to evaluate radical expressions. Practice solving equations with both positive and negative square roots.

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Solving Quadratic Equations using Square Roots

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  1. Do Now:Solve the following equations:x2 = 25x2 = 50Explain your thinking.

  2. Solving Quadratic Equations by Finding Square Roots March 5, 2015

  3. Square Root of a Number If b2= a, then b isthesquareroot of a. Example: If 32 = 9, then 3 isthesquareroot of 9.

  4. VocabularyV All positive real numbers have 2 square roots – Positive square root – principle square root Negative square root Square roots are written with a radical symbol √ Radicand – number inside the radical symbol

  5. Radical Radicand Radical Sign

  6. Positive or Negative • To indicate that we want both the positive and the negative square root of a radicand we put the symbol ± (read as plus minus) in front of the root.

  7. What about zero? • Zero has one square root which is 0. • Negative numbers don't have real square roots since a square is either positive or 0. • The square roots of negative numbers are imaginary numbers. Example : √-9

  8. A negative outside the Radicand A negative sign outside the radicand symbolizes the inverse of the square root. Example: -√9 = -3

  9. Evaluate the expression • √64 • -√64 • √0 • ±√0.25 • √-4

  10. Which of the following are not perfect squares? • -√121 • -√1.44 • √0.09 • √7 √7 is the only irrational number

  11. Radical Expressions The square root symbol is a grouping symbol. Evaluate √b2 -4ac when a=1, b=-2, and c=-3

  12. Solving x2= d If d > 0, then x2 = d has 2 solution: + and – If d = 0, then x2 = d has 1 solution: 0 If d < 0, then x2 = d has no real solution.

  13. Solve each equation • x2 = 2 • x2 = 5 • x2 = -1

  14. Rewriting before finding square roots 3x2 – 48 = 0 3x2 = 48 X2 = 16 X = ±√16 X = ±4

  15. Falling Objects Model h = -16t2 + s h is height in feet t is time in seconds s is the initial height the object was dropped

  16. Solve the Equation If an object is dropped from an initial height 48 feet, how long will it take to reach the ground? h = -16t2 + s 0 = -16t2 + 48 -48 = -16t2 3 = t2 About 1.7 seconds = t

  17. Properties of Square Rootsp Product Property – Example: Quotient Property -

  18. Examples 1. √500 2.

  19. Rationalizing the Denominator You CANNOT leave a radical in the denominator of a fraction! (the numerator is OK) Just multiply the top & bottom of the fraction by the radical to “rationalize” the denominator.

  20. An Example

  21. Try these on your own Solve. • 3 - 5x2 = -9 • 3(x-2)2=21

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