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Completing the Square and Solving Quadratics by Taking Square Roots

Completing the Square and Solving Quadratics by Taking Square Roots. Perfect Square Trinomials. Some quadratics have special factoring rules which we have not yet discussed These are in the form a 2 -2ab+b 2 or a 2 +2ab+b 2 They can be simplified to (a-b) 2 and ( a+b ) 2

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Completing the Square and Solving Quadratics by Taking Square Roots

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  1. Completing the Square and Solving Quadratics by Taking Square Roots

  2. Perfect Square Trinomials • Some quadratics have special factoring rules which we have not yet discussed • These are in the form a2-2ab+b2 or a2+2ab+b2 • They can be simplified to (a-b)2 and (a+b)2 • They can also be simplified using the box method to get the same result

  3. Factor • x2-10x+25 • Matches which form? a2-2ab+b2 or a2+2ab+b2 • Simplified form (a-b)2 and (a+b)2

  4. Factor • x2+8x+16

  5. Factor • 40x=8x2+50

  6. Why bother discussing perfect square trinomials now? • They show up in solving square roots! • They show up when we complete the square!

  7. Solving by taking square roots • To solve a quadratic equation you can take the square root of both sides • First simplify so the squared term is on one side of the equal sign and everything else is on the other! • The key is to remember the positive and negative square roots

  8. Solve 3x2-4

  9. Solve 4x2-20=5

  10. Solve x2-10x+25=27

  11. Solve x2+8x+16=49

  12. Completing the Square • Solving with the method above, can be used only when the expressions work out to be nice squares • When you are presented with a problem not modeled in the form x2+bx, you can add a term to form a perfect square trinomial • Called completing the square

  13. Completing the Square • To complete the square of x2+bx add

  14. Example • x2-2x + __________ • x2+4x+ __________ • x2+3x+ __________ • x2+5x+ __________ • x2-4x + __________

  15. Steps in Completing the Square • 1. Collect variable terms on one side of the equation and constants on the other • 2. As needed, divide both sides by “a” to make the coefficient of the x2term 2 • 3. Complete the square by adding (b/2)2 to EACH side of the equation • 4. Factor the variable as a perfect square trinomial • 5. Take square root of both sides of the equation and solve for the values of the variable including + and -

  16. Solve X2=27-6x • 1. Get all variables to one side • 2. Setup to complete the square • 3. Add (b/2)2 to each side

  17. 4. Perfect square trinomial form • 5. Solve

  18. Solve 2x2+8x=12 • 1. Get all variables to one side • 2. Setup to complete the square • 3. Add (b/2)2 to each side

  19. 4. Perfect square trinomial form • 5. Solve

  20. Solve x2-6x=-4 • 1. Get all variables to one side • 2. Setup to complete the square • 3. Add (b/2)2 to each side

  21. Try in your groups • 3x2-24x=27

  22. 4. Perfect square trinomial form • 5. Solve

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