Download presentation

Presentation is loading. Please wait.

Published byLaurence Hawkins Modified over 6 years ago

1
Completing the square Solving quadratic equations 1. Express the followings in completed square form and hence solve the equations x 2 + 4x – 12 = 0 (x + 2) 2 – 16 = 0 (x + 2) 2 = 16 x + 2 = 16 x + 2 = 4 x = - 2 4 x = -6 or x = 2 = (x + 2) 2 – 2 2 – 12 = 0 2. x 2 + 6x + 4 = 0 (x + 2) 2 – 5 = 0 (x + 3) 2 = 5 x + 3 = 5 x = - 3 5 x = - 3 - 5 or - 3 + 5 = (x + 3) 2 – 3 2 + 4 = 0

2
Sketching graph Express x 2 - 4x -5 in the form (x + p) 2 + q, hence: i) find the minimum value of the expression y = x 2 - 4x - 5. ii) solve the equation x 2 - 4x - 5 = 0 iii) sketch the graph of the function y = x 2 - 4x - 5 Completed square form x 2 – 4x – 5 = y x Vertex (2, -9) The curve is symmetrical about x = 2 (x – 2) 2 – 9 x 2 – 4x – 5 = (x – 2) 2 – 9 = 0 x – 2 = 9 x – 2 = 3 x = 2 3 x = -1 or x = 5 (-1, 0) (5, 0) (x – 2) 2 – 4 - 5 = Solving: x 2 – 4x – 5 = 0 (x – 2) 2 = 9

3
Sketching graph Write 1 + 4x - x 2 in completed square form, hence solve 1 + 4x – x 2 = 0 and sketch the graph of y = 1 + 4x – x 2. Completed square form 1 + 4x – x 2 = - [ x 2 – 4x ] + 1 y x Vertex (2, 5) The curve is symmetrical about x = 2 -[ x 2 – 4x ] + 1 = - [ (x – 2) 2 – 4 ] + 1= - (x – 2) 2 + 4 + 1 = - (x – 2) 2 + 5 - (x – 2) 2 + 5 = 0 - (x – 2) 2 = - 5 (x – 2) 2 = 5 x – 2 = 5 x = 2 5 x = 2 - 5 or x = 2 + 5 (2 - 5) (2 + 5)

4
Sketching graph Write -3x 2 + 6x - 2 in completed square form, hence solve - 3x 2 + 6x – 2 and sketch the graph of y = -3x 2 + 6x – 2. Completed square form -3[ x 2 - 2x ] – 2 = -3[ (x - 1) 2 - 1 ] - 2 y x Vertex ( 1, 1 ) The curve is symmetrical about x = 1 = -3(x - 1) 2 + 3 - 2 = -3(x - 1) 2 + 1 -3(x - 1) 2 + 1 = 0 -3(x - 1) 2 = - 1 (1 - (1/3), 0) (1 + (1/3), 0)

5
More examples Complete the square for each of the following quadratic functions and solve f(x) = 0 (a)x 2 + x – ½ =(x + ½ ) 2 – ¼ – ½ =(x + ½ ) 2 – ¾ (c)3 + 4x – 2x 2 =-2 [x 2 + 2 x ] + 3 = 2[(x + 1 ) 2 – 1 ] + 3 = 2(x + 1 ) 2 – 2 + 3 = 2(x + 1 ) 2 + 1 (x + ½ ) 2 – ¾ = 0 (x + ½ ) 2 = ¾ x + ½ = ¾ x = -½ ¾ 2(x + 1 ) 2 + 1 = 0 2(x + 1 ) 2 = - 1 (x + 1 ) 2 = - ½ No solution

6
The function f is defined for all x by f(x) = x 2 + 3x – 5. a) Express f(x) in the form (x + P) 2 + Q. Complete the square

7
Solve the equation f(x) = 0 by making x the subject, using the completed square format b) Hence, or otherwise, solve the equation f(x) = 0, giving your answers in surd form. Tip: You could have used the quadratic formula on x 2 + 3x – 5 = 0. Tip: Simplify the surd where

Similar presentations

© 2021 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google