5.5: Completing the Square Objectives: Students will be able to… Solve a quadratic equation by completing the square Use completing the square to write quadratic functions in vertex form.

Completing the Square Allows you to write a quadratic expression x2 + bx as the square of a binomial Examples of squares of binimials: (x + 3)2 (x – 6)2

To Complete the Square: x2 + bx + ____ Divide b by 2: Square this quantity: Add to the end of x2 + bx: Now you can write it as the square of a binomial:

Examples: Find the values of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.

Solving a Quadratic by Completing the Square when a = 1 x2 + 6x -8 =0 Isolate your x terms: Complete the square; add quantity to BOTH sides of equation: Write the left side as a binomial squared: Solve by taking the square root of both sides:

Solve by Completing the Square 1. x2+4x-1=0 2. x2-3x=1

Solve by completing the square when a ≠ 1 You must divide ENTIRE equation by a before you begin. Then proceed as when a = 1. 5x2 -10x + 30= 0 Divide every term by 5:

Solve by Completing the Square 1. 4x2 – 4x -2 = 0 2. -x2 -4x +3 =0

Using Completing the Square to rewrite a Quadratic function in vertex form. Write the quadratic in vertex form. What is the vertex of the function’s graph? y = x2 + 6x + 16 Separate the x terms on right side: Complete the square and add this quantity to both sides of function: Write the quadratic as the square of a binomial: Solve for y:

Write in vertex form. y = x2 – 2x - 9

If a ≠ 1: y = 3x2 – 12x +1 Before you begin, divide a out of x terms ONLY: Complete the square; when adding quantity to left side, add : Rest of the problem is the same:

Write in vertex form. What is the vertex? y = -2x2 – 2x - 7

Under certain road conditions, the formula for a car’s stopping distance is given by d = 0.1s2 +1.1s If a driver leaves 5 car lengths, approximately 75 feet, between him and the driver in front of him, what is the maximum speed he can drive and still stop safely?

You have 30 feet of chain link fence to make a rectangular enclosure for your dog. A pet store owner recommended that an enclosure for one dog be at least 48 ft2 in area. What should the dimensions of the enclosure be to make the area 48 ft2?

You are making a fence to enclose a rose garden on the side of your garden shed. You have 10 ft. of fence and want to make the garden have an area of 9 ft2. What are the possible dimensions?

The number of people n who attend a traveling circus can be modeled by the function where x is the number of times the circus was advertised on the local radio station during the month prior to the show. How many times should the circus advertise to maximize the number of people attending? What is the maximum number of people?

Tying it all together… Solve the quadratic equation using factoring, square roots, or completing the square. 2x2 + 9x + 7 =0 4(x +8)2 = 144 X2 – 6x -15 =0