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Completing the Square CCSS: A.SSE.3; F.IF.7

Completing the Square CCSS: A.SSE.3; F.IF.7. Solving Quadratics By Completing the Square. Must be a perfect Square. CCSS: A.SSE.3. CHOOSE and PRODUCE an equivalent form of an expression to REVEAL and EXPLAIN properties of the quantity represented by the expression.

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Completing the Square CCSS: A.SSE.3; F.IF.7

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  1. Completing the SquareCCSS: A.SSE.3; F.IF.7 Solving Quadratics By Completing the Square Must be a perfect Square

  2. CCSS: A.SSE.3 • CHOOSE and PRODUCE an equivalent form of an expression to REVEAL and EXPLAIN properties of the quantity represented by the expression. • a. FACTOR a quadratic expression to reveal the zeros of the function it defines. • b. COMPLETE THE SQUARE iin a quadratic expression to REVEAL the maximum or minimum value of the function it defines.

  3. CCSS: F.IF.7 • GRAPH functions expressed symbolically and SHOW key features of the graph, by hand in simple cases and using technology for more complicated cases.* • a. GRAPH linear and quadratic functions and show intercepts, maxima, and minima.

  4. Standards for Mathematical Practice • 1. Make sense of problems and persevere in solving them. • 2.Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the reasoning of others. • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6.Attend to precision. • 7. Look for and make use of structure. • 8.Look for and express regularity in repeated reasoning.

  5. Essential Questions • How do I determine the domain, range, maximum, minimum, roots, and y-intercept of a quadratic function from its graph? • How do I use quadratic functions to model data?  • How do I solve a quadratic equation with non - real roots?

  6. Perfect Square On One side Review Solve for x Take Square Root of BOTH SIDES When you take the square root, You MUST consider the Positive and Negative answers.

  7. Perfect Square On One side Solve for x Take Square Root of BOTH SIDES But what happens if you DON’T have a perfect square on one side……. You make it a Perfect Square Use the relations on next slide…

  8. Short Cut To expand a perfect square binomial: We can use these relations to find the missing term….To make it a perfect square trinomial that can be factored into a perfect square binomial. Then

  9. Make this a perfect square trinomial • Take ½ middle term • Then square it • The resulting trinomial is called a perfect square trinomial, • which can be factored into a perfect square binomial.

  10. Make one side a perfect square • Add a blank to both sides • Divide “b” by 2 • 4. Square that answer. • Add it to both sides • Factor 1st side • Square root both sides • Solve for x Solve by completing the square 1.

  11. Short cut! Factor this Perfect square trinomial Bring down sign What is the Square root of x2 What is the Square root of 36

  12. Move constant to other side. • Add a blank to both sides • Divide “b” by 2 • 4. Square that answer. • Add it to both sides • Factor 1st side • Square root both sides • Solve for x Solve by completing the square 2. -8 -8

  13. Short cut! Factor this Perfect square trinomial Bring down sign What is the Square root of x2 What is the Square root of 9

  14. Move constant to other side. • Add a blank to both sides • Divide “b” by 2 • Square that answer. • Add it to both sides • Factor 1st side • Square root both sides • Solve for x Solve by completing the square 3. +84 +84

  15. Short cut! Factor this Perfect square trinomial Bring down sign What is the Square root of x2 What is the Square root of 9

  16. Move constant to other side. • Add a blank to both sides • Divide “b” by 2 • Square that answer. • Add it to both sides • Factor 1st side • Square root both sides • Solve for x Solve by completing the square 4. +15 +15

  17. Short cut! Factor this Perfect square trinomial Bring down sign What is the Square root of x2 What is the Square root of 9

  18. Move the constant to side by itself. Make the side (w/variables) a perfect square by adding a certain number to both sides. To calculate this number Divide “b” (middle term) by 2 Then square that answer Take the square root of both sides of eq Then solve for x Steps to solve Quadratics by completing the square:

  19. In a perfect square, there is a relationship between the coefficient of the middle term and the constant term.

  20. Creating a Perfect Square Trinomial • In the following perfect square trinomial, the constant term is missing. X2 + 14x + ____ • Find the constant term by squaring half the coefficient of the linear term. • (14/2)2 X2 + 14x + 49

  21. Perfect Square Trinomials • Create perfect square trinomials. • x2 + 20x + ___ • x2 - 4x + ___ • x2 + 5x + ___ 100 4 25/4

  22. Solving Quadratic Equations by Completing the Square Try the following examples. Do your work on your paper and then check your answers.

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