Derive the Quadratic Formula

Derive the Quadratic Formula

Derive the Quadratic Formula Focus 10 Learning Goal (HS.A-REI.B.4, HS.F-IF.B.4, HS.F-IF.C.7, HS.F-IF.C.8) = Students will sketch graphs of quadratics using key features and solve quadratics using the quadratic formula. 4 3 2 1 In addition to level 3, students

make connections to other content areas and/or contextual situations outside of math. Students will sketch graphs of quadratics using key features and solve quadratics using the quadratic formula. - Students will be able to write, interpret and graph quadratics in vertex form.

Students will be able to use the quadratic formula to solve quadratics and are able to identify some key features of a graph of a quadratic. Students will have partial success at a 2 or 3, with help. 0 Even with help, the student is

not successful at the learning goal. Solve for x by completing the square: x2 + 2x 8 = 0 1. Add 8 to both sides: 1. x2 + 2x + ___ = 8 2. (1/2b)2 = (1/2 2)2 = 1 3. Add 1 to both sides. 1. x2 + 2x + 1 = 9 4. Factor: 1. (x + 1)2 = 9 5. Square root both sides: 1. x + 1 = 3, -3 6. Get x alone: 1. x = 2 and -4

7. Note: This equation could have been solved by factoring. Solve for x by completing the square: 7x2 - 12x + 4 = 0 1. Divide all terms by the leading coefficient. 1. 7/7x2 5. Factor: 1. (x - 6/7)2 = 8/49 /7x + 4/7 = 0 12 2. Subtract /7 on both sides: 4 1. x2 -

/7x + ___ = - 4/7 12 3. (1/2b)2 = (1/2 4. Add /7)2 = 12 /49 36 /49 to both sides. 36 1. x2 - 12

/7x + 36 2. x2 - 12 /49 = - 4/7 + /7x + 36 /49 = 8/49 /49 36 6.Square root both sides: 1. x - 6/7 =

7.Get x alone: 1. x = Reflection: Which of these two problems makes more sense to solve by completing the square? Which makes more sense to solve by factoring? How could you tell early in the problem solving process which strategy to use? How would you How would you solve this equation for x: ax + b = 0 (where a & b could be replaced by any number)? Subtract b: ax = -b Divide by a: x = -b/a We could say that b/a is a formula to solve ANY equation in the form ax + b = 0. Try it: 2x + 7 = 0 x = -7/2

How would you How would you solve this equation for x: ax2 + bx + c = 0 (where a, b & c could be replaced by any number)? Since we dont know if this will factor and leave rational answers, we could use the method of completing the square to get x alone. What would this look like since we dont know the value of a? The Quadratic Formula! You will need to memorize this! Dont worry, there is a catchy song to help you with this ! Singing Time! Practice using the quadratic formula instead of factoring or completing the square to solve x2 + 5x 14 = 0.

2 (5) 5 41 14 x 2 1 5 25 56 2 59 2 2 5 81 2 5 9 2

x = 2,-7 5 9 2 7 Practice using the quadratic formula instead of factoring or completing the square to solve x2 + 2x 5 = 0. 2 2 2 4(1)( 5) 2(1) 2 4 20 2 22 6

2 2 2 24 AND 2 2 6 2 Solve these quadratic equations, using a different method for each: solve by factoring, solve by completing the square and solve by using the quadratic formula. Before starting, decide which method you will use on each one. 1. 2x2 + 5x 3 =0 This one could be factored to solve.

x = or -3 2. x2 + 3x 5 = 0 3. x2 - x 4 = 0 This one could have been solved by using the quadratic formula or by completing the square. This one could have been factored to solve. Because of the leading coefficient, you may have tried a different strategy. x= x = 4 or -2

When do you factor, complete the square, or use the quadratic formula? When is completing the square the most efficient method to use for solving a quadratic equation? When it is not possible (or it is very difficult) to factor the quadratic expression and when the leading coefficient and linear term are easy to deal with. When is the quadratic formula best? When it is not possible (or it is very difficult) to factor the quadratic expression, and when the leading coefficient and/or linear term coefficient are fractions that are not easily eliminated. When is factoring the most efficient method to use for solving a quadratic equation? When the factors of the equation are obvious or fairly easy to find. When factoring out the GCF or eliminating any fractional coefficients is possible.

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