Presentation on theme: "Section 9.1 MATH Mr. Keltner"— Presentation transcript:
1 Section 9.1 MATH 116-460 Mr. Keltner Completing the SquareSection 9.1MATHMr. Keltner
2 Example 1Solve (x + 6)2 = 20.We know how to solve equations when it is possible to take the square root of each side, such as:4x2 = 16 Constant multiple with a variablex2 + 14x + 49 = 25 A quantity that can be rewrittenx2 = -76 Where we take the square root of a negative number, or use i
3 A need for a new strategy Sometimes, we will have an expression of the form x2 + bx, that requires us to insert an additional term to make it a perfect square, like (x + k)2.The process of forcing a quadratic expression to become a perfect square trinomial is called completing the square.
4 Completing the Square in 3 “Easy” Steps These steps may only be used on a quadratic expression in the form ax2 + bx, where a = 1, and b is a real number.Find half the coefficient of x.Square that value.Replace c with the resulting value.Example:Find the value of c that makes x2 -26x + c a perfect square trinomial. Then write the expression as the square of a trinomial.-26 ÷ 2 = -13(-13)2 = 169x2 -26x + c becomes x2 -26x + 169, which is equal to (x-13)2 when it is factored.
5 Example 2Find the value of c that makes each expression a perfect square trinomial. Then write each expression as the square of a binomial.x2 + 14x + cx2 + 22x + cx2 – 9x + cWhat do you notice about the value of c if b happens to be an odd number?Does the sign of c change or does it remain constant?
6 Solving Quadratic Equations by Completing the Square Related example:Solve x2 - 4 = 12.Your first step would be to add 4 to each side of the equation.Just the same as we add the same value to both sides of this equation, we apply this same idea when completing the square.Example 3Solve x2 - 10x + 1 = 0 by completing the square.
7 Solving ax2 + bx + c = 0 if a ≠ 1Divide every term of both sides by the coefficient of x2 (the value of a).Make sure to balance the equation by adding the same value to both sides.Example 4: Solve 3x2 – 36x = 0 by completing the square.
8 Example 5 Solve the equation by completing the square. 6x (x + 8) = 12 (NOTE: You cannot apply the Zero-Product Property and say that either 6x = 12 or (x + 8) = 12 and solve.)6x (x + 8) = 12
9 Baseball example: Finding a maximum value The height, y (in feet), of a baseball x seconds after it is hit is given by the equation:y = -16x2 + 96x + 3Find the maximum height of the baseball.The maximum height of the baseball will be the y-coordinate of the vertex of the parabola.It will help if we can write the equation in vertex form.
10 Baseball Example, Cont.The height, y (in feet), of a baseball x seconds after it is hit is given by the equation:y = -16x2 + 96x + 3Find the maximum height of the baseball.Start by writing the function in vertex form.If we can find the y-value at the vertex, we will have found the maximum height of the ball.
11 Fountain ExampleAt the Buckingham Fountain in Chicago, the water’s height h (in feet) above the main nozzle can be modeled by h = -16t t, where t is the time in seconds) since the water has left the nozzle.Find the highest point the water reaches above the fountain.What does this vertex represent, in real-world terms?
12 Pgs. 628-631: #’s 7-98, multiples of 7 AssessmentPgs :#’s 7-98, multiples of 7